Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y \in X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f(Y) \in Y$~$ |
---|

Axiom of Choice: Definition (Formal) | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y_1, Y_2, Y_3$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f(Y_1) = y_1$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f(Y_2) = y_2$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f(Y_3) = y_3$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$n$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$n$~$ |
---|

Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|

""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|

""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|

""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$-\infty$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$+\infty,$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$0$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$1$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$0$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$1$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$\mathbb P(X) + \mathbb P(\lnot X)$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$\lnot X$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$X$~$ |
---|

""Extreme credences" here should likely be "infi..." | $~$\aleph_0$~$ |
---|

""Formula" and "Statement" can be interchanged f..." | $~$\{+,\dot,0,1\}$~$ |
---|

""That's because we're considering results like ..." | $~$2^6 = 64$~$ |
---|

""That's because we're considering results like ..." | $~$p<0.05$~$ |
---|

""We only ran the 2012 US Presidential Election ..." | $~$10 bet that paid out $~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$8$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$4$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$\log_4 8$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$1.5$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$3$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$2$~$ |
---|

"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$log_2 3$~$ |
---|

"(5) was intended to assume that $n \in \mathbb ..." | $~$n \in \mathbb R^{\ge 1},$~$ |
---|

"(5) was intended to assume that $n \in \mathbb ..." | $~$\in \mathbb R^{\ge 0}$~$ |
---|

"(5) was intended to assume that $n \in \mathbb ..." | $~$f(x^y)=yf(x)$~$ |
---|

"(5) was intended to assume that $n \in \mathbb ..." | $~$f(b^n)=nf(b)$~$ |
---|

"(5) was intended to assume that $n \in \mathbb ..." | $~$f(b)=1 \implies f(b^n)=n,$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$n$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$f$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$(\mathbb{R}^{>0},\cdot)$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$(\mathbb{R},+)$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$log$~$ |
---|

"(8) doesn't follow from (5). The assumption in ..." | $~$\mathbb{R}$~$ |
---|

"1. I propose that this concept be called "unex..." | $$~$ s(d) = \textrm{surprise}(d \mid H) = - \log \Pr (d \mid H) $~$$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$(d \mid H)$~$ |
---|

"1. I propose that this concept be called "unex..." | $$~$\textrm{log-likelihood} = -\textrm{surprise}$~$$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$t(d)$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$\Pr(d \mid H)$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $$~$\Pr(H \mid d) = \Pr(H \mid t(d))$~$$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|

"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|

"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(e \mid GoodDriver)$~$ |
---|

"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(e \mid BadDriver)$~$ |
---|

"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(BadDriver)$~$ |
---|

"A summary of the relevant cardinal arithmetic, ..." | $$~$\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} = \aleph_{\alpha} \aleph_{\alpha}$~$$ |
---|

"A summary of the relevant cardinal arithmetic, ..." | $$~$2^{\aleph_{\alpha}} > \aleph_{\alpha}$~$$ |
---|

"Actually, there should be diagonal matrices ins..." | $~$\mathbf H$~$ |
---|

"Actually, there should be diagonal matrices ins..." | $~$H_1, H_2, \ldots$~$ |
---|

"Actually, there should be diagonal matrices ins..." | $~$\mathbf H,$~$ |
---|

"Actually, there should be diagonal matrices ins..." | $~$C = AB; c_{ii} = a_{ii} * b_{ii}; ∀ i ≠ j, c_{ij} = 0$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_i$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$x_i$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_i$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_0$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_1$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_2$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$X_3$~$ |
---|

"Ah, one additional thing I'm confused about -- ..." | $~$x_i$~$ |
---|

"Another, speculative point:
If $V$ and $U$ we..." | $~$V$~$ |
---|

"Another, speculative point:
If $V$ and $U$ we..." | $~$U$~$ |
---|

"Any relation satisfying 1-3 is a partial order,..." | $~$S$~$ |
---|

"Any relation satisfying 1-3 is a partial order,..." | $~$\le$~$ |
---|

"Are all the words in the free group, or just th..." | $~$X$~$ |
---|

"Are all the words in the free group, or just th..." | $~$X \cup X^{-1}$~$ |
---|

"Are all the words in the free group, or just th..." | $~$r r^{-1}$~$ |
---|

"Are all the words in the free group, or just th..." | $~$r^{-1} r$~$ |
---|

"Are all the words in the free group, or just th..." | $~$r r^{-1}$~$ |
---|

"Are all the words in the free group, or just th..." | $~$r \in X$~$ |
---|

"Are all the words in the free group, or just th..." | $~$r^{-1} r$~$ |
---|

"Are all the words in the free group, or just th..." | $~$X \cup X^{-1}$~$ |
---|

"Be wary here.
We see on the next (log probabil..." | $~$(1 : 10^{100})$~$ |
---|

"Be wary here.
We see on the next (log probabil..." | $~$(1 : 10^6)$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$n$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x \cdot x \le n$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x=316$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|

"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x^2 \le 100000.$~$ |
---|

"Broken link :(" | $~$M$~$ |
---|

"Broken link :(" | $~$N$~$ |
---|

"Consider using [3jp] for the proof?" | $~$x!$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$x! = \Gamma (x+1),$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$\Gamma $~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$x$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$x=1$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{1}i = \int_{0}^{\infty}t^{1}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$1=1$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$x$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$x + 1$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x+1}i = \int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $~$x+1$~$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$(x+1)\prod_{i=1}^{x}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x+1}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$= 0+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}-\int_{0}^{\infty}(x+1)t^{x}(-e^{-t})\mathrm{d} t$~$$ |
---|

"Consider using [3jp] for the proof?" | $$~$=\int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$(S, \le)$~$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$S$~$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\le$~$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$S$~$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\leq$~$ |
---|

"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\leq$~$ |
---|

"Darn it, I wanted to use th..." | $~$Y$~$ |
---|

"Darn it, I wanted to use th..." | $~$X$~$ |
---|

"Darn it, I wanted to use th..." | $~$X$~$ |
---|

"Darn it, I wanted to use th..." | $~$Y$~$ |
---|

"Darn it, I wanted to use th..." | $~$X$~$ |
---|

"Darn it, I wanted to use th..." | $~$Y$~$ |
---|

"Darn it, I wanted to use th..." | $~$X$~$ |
---|

"Darn it, I wanted to use th..." | $~$X.$~$ |
---|

"Do the different biases of coin correspond to d..." | $~$H_{0.55},$~$ |
---|

"Do the different biases of coin correspond to d..." | $~$H_{0.6}$~$ |
---|

"Do the different biases of coin correspond to d..." | $~$H_{0.8}.$~$ |
---|

"Do the different biases of coin correspond to d..." | $~$H_{0.5},$~$ |
---|

"Does this actually work for..." | $~$A$~$ |
---|

"Does this actually work for..." | $~$B$~$ |
---|

"Does this actually work for..." | $~$\bP$~$ |
---|

"Does this actually work for..." | $~$\bP$~$ |
---|

"Does this make the definiti..." | $~$Y$~$ |
---|

"Does this make the definiti..." | $~$f$~$ |
---|

"Does this make the definiti..." | $~$Y$~$ |
---|

"Does this make the definiti..." | $~$\operatorname{square} : \mathbb R \to \mathbb R$~$ |
---|

"Does this make the definiti..." | $~$\operatorname{square}(x)=x^2$~$ |
---|

"Does this make the definiti..." | $~$\mathbb R$~$ |
---|

"Does this make the definiti..." | $~$\mathbb R$~$ |
---|

"Does this make the definiti..." | $~$\mathbb R$~$ |
---|

"Does this make the definiti..." | $~$\mathbb C$~$ |
---|

"Does x correspond to a *statement* (as used in ..." | $~$Prv(x)$~$ |
---|

"Does x correspond to a *statement* (as used in ..." | $~$x$~$ |
---|

"For readers who just skimme..." | $~$n$~$ |
---|

"For readers who just skimme..." | $~$2^n$~$ |
---|

"For readers who just skimme..." | $~$2^{3,000,000,000,000}$~$ |
---|

"For readers who just skimme..." | $~$2^{3,000,000,000,000}$~$ |
---|

"For readers who just skimme..." | $~$2^\text{3 trillion}$~$ |
---|

"Had to re-read this twice. ..." | $~$a$~$ |
---|

"Had to re-read this twice. ..." | $~$b$~$ |
---|

"Had to re-read this twice. ..." | $~$31a + b$~$ |
---|

"Had to re-read this twice. ..." | $~$31\cdot 30 + 30 = 960$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$\mathcal L(H \mid e) < 0.05$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$\mathcal L(H \mid e)$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|

"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$b$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$n,$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$\log_b(n),$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$b$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$n$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$\log_{10}(100)=2,$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$\log_{10}(316) \approx 2.5,$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$316 \approx$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$10 \cdot 10 \cdot \sqrt{10},$~$ |
---|

"Having a long redlink which does not point anyw..." | $~$\sqrt{10}$~$ |
---|

"How about, "because I'm goi..." | $~$\log_{10}(\text{2,310,426})$~$ |
---|

"Huh... Not sure I understand this. I have BS in..." | $~$f$~$ |
---|

"Huh... Not sure I understand this. I have BS in..." | $~$x$~$ |
---|

"Huh... Not sure I understand this. I have BS in..." | $~$f(x)$~$ |
---|

"Huh... Not sure I understand this. I have BS in..." | $~$1/2$~$ |
---|

"Huh... Not sure I understand this. I have BS in..." | $~$f$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{fair},$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{heads}$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{tails}$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(1/2 : 1/3 : 1/6).$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(3 : 2 : 1)$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(2 : 1 : 3).$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(2 : 3 : 1)$~$ |
---|

"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(3 : 2 : 1)$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$A$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$B$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$C$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$C$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$\mathcal T$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$B$~$ |
---|

"I can't figure out what this paragraph means --..." | $~$D$~$ |
---|

"I don't think this is what you mean, is it?" | $~$X$~$ |
---|

"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|

"I don't think this is what you mean, is it?" | $~$X$~$ |
---|

"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|

"I don't think this is what you mean, is it?" | $~$X \to Y$~$ |
---|

"I don't think this is what you mean, is it?" | $~$Y^X$~$ |
---|

"I don't think this is what you mean, is it?" | $~$Y^2$~$ |
---|

"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|

"I don't understand this sen..." | $$~$1$~$$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$99\cdot 2=198$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$100$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$LDT$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$198$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$100$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$1$~$ |
---|

"I fail to see how this setup is not fair - but ..." | $~$0$~$ |
---|

"I got lost here (and in the following equations..." | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|

"I got lost here (and in the following equations..." | $~$X_i$~$ |
---|

"I got lost here (and in the following equations..." | $~$x_i$~$ |
---|

"I got lost here (and in the following equations..." | $~$\mathbf {pa}_i$~$ |
---|

"I got lost here (and in the following equations..." | $~$x_i$~$ |
---|

"I got lost here (and in the following equations..." | $~$\mathbf x$~$ |
---|

"I got lost here -- I feel l..." | $~$\bullet$~$ |
---|

"I got lost here -- I feel l..." | $~$G$~$ |
---|

"I got lost here -- I feel l..." | $~$G$~$ |
---|

"I love the effect, but I wo..." | $~$t = 0$~$ |
---|

"I love the effect, but I wo..." | $~$4.7 t^2$~$ |
---|

"I love the effect, but I wo..." | $~$t$~$ |
---|

"I might write this as, "whe..." | $~$x$~$ |
---|

"I might write this as, "whe..." | $~$n$~$ |
---|

"I might write this as, "whe..." | $~$n-1$~$ |
---|

"I might write this as, "whe..." | $~$n$~$ |
---|

"I might write this as, "whe..." | $~$\log_{10}(x)$~$ |
---|

"I might write this as, "whe..." | $~$x;$~$ |
---|

"I might write this as, "whe..." | $~$x$~$ |
---|

"I might write this as, "whe..." | $~$x$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(n)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$n$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(n)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$n$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(m)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$k \geq m$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(k)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(k+1)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(m)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(m+1)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(m+1)$~$ |
---|

"I really like this domino analogy.
Also, I'd e..." | $~$P(m+2)$~$ |
---|

"I see that there is a description of double sca..." | $~$-1$~$ |
---|

"I suggest making it explici..." | $~$P$~$ |
---|

"I suggest making it explici..." | $~$P(x)$~$ |
---|

"I suggest making it explici..." | $~$P(X=x)$~$ |
---|

"I suggest making it explici..." | $~$X$~$ |
---|

"I suggest making it explici..." | $~$P$~$ |
---|

"I suggest we can assume tha..." | $~$s$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$n \ge 1$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$n=1$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$ 1 = \frac{1(1+1)}{2} = \frac{2}{2} = 1.$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$k$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$k\ge1$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$1 + 2 + \cdots + k = \frac{k(k+1)}{2}$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$k+1$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$1+2+\cdots + k + (k+1) = \frac{k(k+1)}{2} + k + 1.$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$\frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{(k+2)(k+1)}{2} = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}$~$$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$n$~$ |
---|

"I think it would be worthwhile to explicitly ca..." | $~$k+1$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda x.f(x)$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$x$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$f(x)$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda x.x+1$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda x.\lambda y.x+y$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda xy.x+y$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda xy$~$ |
---|

"I think it's confusing to introduce multi-argum..." | $~$\lambda x.\lambda y$~$ |
---|

"I think that every metric space is dense in its..." | $~$\newcommand{\rats}{\mathbb{Q}} \newcommand{\Ql}{\rats^\le} \newcommand{\Qr}{\rats^\ge} \newcommand{\Qls}{\rats^<} \newcommand{\Qrs}{\rats^>}$~$ |
---|

"I think that every metric space is dense in its..." | $~$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sothat}{\ |\ }$~$ |
---|

"I think the answer is no. Indeed, there are unc..." | $~$S$~$ |
---|

"I think this paragraph and ..." | $~$2^6 < 101 < 2^7$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|

"I think this sentence would be easier to read w..." | $~$(\lambda x.(\lambda y.(x+y)))$~$ |
---|

"I think this sentence would be easier to read w..." | $~$f\ x\ y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$f$~$ |
---|

"I think this sentence would be easier to read w..." | $~$x$~$ |
---|

"I think this sentence would be easier to read w..." | $~$y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$(f\ x)\ y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$f\ (x\ y)$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda x.\lambda y.x+y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|

"I think this sentence would be easier to read w..." | $~$(\lambda x.\lambda y.x)+y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.x)+y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda xy.x+y$~$ |
---|

"I think this sentence would be easier to read w..." | $~$\lambda x.\lambda y.x+y$~$ |
---|

"I think you may need to spe..." | $~$x$~$ |
---|

"I think you may need to spe..." | $~$n$~$ |
---|

"I think you may need to spe..." | $~$n-1$~$ |
---|

"I think you may need to spe..." | $~$n$~$ |
---|

"I think you may need to spe..." | $~$\log_{10}(x)$~$ |
---|

"I think you may need to spe..." | $~$x;$~$ |
---|

"I think you may need to spe..." | $~$x$~$ |
---|

"I think you may need to spe..." | $~$x$~$ |
---|

"I would consider leading wi..." | $~$\log_{10}(12) \approx 1.08$~$ |
---|

"I would consider leading wi..." | $~$\log_2(10) \approx 3.32$~$ |
---|

"I would expect this sentence only after another..." | $~$2 : 1$~$ |
---|

"I would expect this sentence only after another..." | $~$8 : 1,$~$ |
---|

"I would expect this sentence only after another..." | $~$2 : 1$~$ |
---|

"I would expect this sentence only after another..." | $~$4 : 1.$~$ |
---|

"I'm curious if the inverse ..." | $~$(a_1 a_2 \dots a_k)$~$ |
---|

"I'm curious if the inverse ..." | $~$(a_k a_{k-1} \dots a_1)$~$ |
---|

"If these are included I think it would be good ..." | $~$0.999\dotsc=1$~$ |
---|

"If you look on Wikipedia's ..." | $~$A \cdot B = A + A + A$~$ |
---|

"If you look on Wikipedia's ..." | $~$B$~$ |
---|

"If you're going to start us..." | $~$\mathbb P$~$ |
---|

"If you're going to start us..." | $~$\operatorname{d}\!f$~$ |
---|

"If you're going to start us..." | $~$\operatorname{d}\!f$~$ |
---|

"In this page, the terms "probability" and "odds..." | $~$X$~$ |
---|

"In this page, the terms "probability" and "odds..." | $~$\mathbb P(X)$~$ |
---|

"In this page, the terms "probability" and "odds..." | $~$X.$~$ |
---|

"In this sentence I think yo..." | $~$f$~$ |
---|

"In this sentence I think yo..." | $~$X$~$ |
---|

"In this sentence I think yo..." | $~$I$~$ |
---|

"In this sentence I think yo..." | $~$Y$~$ |
---|

"In this sentence I think yo..." | $~$I$~$ |
---|

"Intro should be re-written ..." | $~$(X, \bullet)$~$ |
---|

"Intro should be re-written ..." | $~$X$~$ |
---|

"Intro should be re-written ..." | $~$\bullet$~$ |
---|

"Intro should be re-written ..." | $~$X$~$ |
---|

"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-3}$~$ |
---|

"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-1}$~$ |
---|

"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-4}$~$ |
---|

"Is "-1 against" the same as "+1 for"?
Expressi..." | $~$(1 : 16)$~$ |
---|

"Is $\mathbb{N}$ itself called $\omega$, or just..." | $~$\mathbb{N}$~$ |
---|

"Is $\mathbb{N}$ itself called $\omega$, or just..." | $~$\omega$~$ |
---|

"Is [0, inf) same as R+?" | $~$d$~$ |
---|

"Is [0, inf) same as R+?" | $~$d$~$ |
---|

"Is [0, inf) same as R+?" | $~$S$~$ |
---|

"Is [0, inf) same as R+?" | $$~$d: S \times S \to [0, \infty)$~$$ |
---|

"Is this a typo? Shouldn't you buy coins if they..." | $~$10^{10} < 2^{35}.$~$ |
---|

"Is this paragraph needed? ..." | $~$x$~$ |
---|

"Is this paragraph needed? ..." | $~$n$~$ |
---|

"Is this paragraph needed? ..." | $~$n-1$~$ |
---|

"Is this paragraph needed? ..." | $~$n$~$ |
---|

"Is this paragraph needed? ..." | $~$\log_{10}(x)$~$ |
---|

"Is this paragraph needed? ..." | $~$x;$~$ |
---|

"Is this paragraph needed? ..." | $~$x$~$ |
---|

"Is this paragraph needed? ..." | $~$x$~$ |
---|

"Is this paragraph needed? ..." | $~$x$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$A = \{ \{ 1,2 \}, \{ 3,4 \}, 1, 2, 3, 4 \}$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$x = \{1,2\}$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$a = 2$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$a \in x$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$x \in A$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$a \in A$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$B = \{ \{ 1,2 \}, \{ 3,4 \} \}$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$y = \{1,2\}$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$b = 2$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$b \in y$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$y \in B$~$ |
---|

"Is this what is meant by transitive and nontran..." | $~$b \notin B$~$ |
---|

"Is what follows the colon m..." | $~$3^{10}$~$ |
---|

"Is what follows the colon m..." | $~$n^k$~$ |
---|

"Isn't one coin and three di..." | $~$\log_2(6) + \log_2(10) + 3\log_2(2) \approx 8.9$~$ |
---|

"Isn't one coin and three di..." | $~$2*3^6 = 432,$~$ |
---|

"Isn't one coin and three di..." | $~$\log_2(2) + 3*\log_2(6) \approx 8.75$~$ |
---|

"It is really confusing to apply one of the init..." | $~$\mathbb P({positive}\mid {HIV}) = .997$~$ |
---|

"It is really confusing to apply one of the init..." | $~$\mathbb P({negative}\mid \neg {HIV}) = .998$~$ |
---|

"It is really confusing to apply one of the init..." | $~$\mathbb P({positive} \mid \neg {HIV}) = .002.$~$ |
---|

"It would be nice to show how to go from 99.8% t..." | $~$1 : 100,000$~$ |
---|

"It would be nice to show how to go from 99.8% t..." | $~$500 : 1.$~$ |
---|

"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(sick \mid blackened)$~$ |
---|

"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(sick \wedge blackened)$~$ |
---|

"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(blackened)$~$ |
---|

"Looks like a mathjax error?" | $~$PA$~$ |
---|

"Looks like a mathjax error?" | $~$\square_{PA}$~$ |
---|

"Looks like a mathjax error?" | $~$PA$~$ |
---|

"Looks like a mathjax error?" | $~$PA$~$ |
---|

"Looks like a mathjax error?" | $~$A$~$ |
---|

"Looks like a mathjax error?" | $~$\square_{PA}(\ulcorener A\urcorner$~$ |
---|

"Looks like a mathjax error?" | $~$A$~$ |
---|

"Looks like a mathjax error?" | $~$PA$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$x$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$x$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$n$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$c$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$n$~$ |
---|

"May need to build the intuition that knowing ho..." | $~$c.$~$ |
---|

"Maybe insert an equation style definition of th..." | $~${\bf \hat v}$~$ |
---|

"Maybe insert an equation style definition of th..." | $$~$|\mathbf{\hat v}| = \left|\frac{\mathbf{v}}{|\mathbf{v}|}\right| = \left|\frac{1}{|\mathbf{v}|}\right||\mathbf{v}| = \frac{|\mathbf{v}|}{|\mathbf{v}|}=1$~$$ |
---|

"Maybe insert an equation style definition of th..." | $~$\hat{\mathbf v} = \frac{1}{| \mathbf v |}\mathbf v = \frac{\mathbf v}{| \mathbf v |}$~$ |
---|

"Might one of the following ..." | $~$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$~$ |
---|

"Might one of the following ..." | $~$\frac{1}{2}$~$ |
---|

"Might one of the following ..." | $~$G_0 \xrightarrow{f_1} G_1 \xrightarrow{f_2} G_2 \xrightarrow{f_3} \cdots \xrightarrow{f_n} G_n$~$ |
---|

"Might one of the following ..." | $~$\text{im}(f_k) = \text{ker}(f_{k+1})$~$ |
---|

"Might one of the following ..." | $~$0 \le k < n$~$ |
---|

"Might one of the following ..." | $~$n\times n$~$ |
---|

"Might one of the following ..." | $~$A$~$ |
---|

"Might one of the following ..." | $~$a_{i,j}$~$ |
---|

"Might one of the following ..." | $~$\det(A) = \sum_{\sigma\in S_n}\text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma_i}$~$ |
---|

"Might one of the following ..." | $~$S_n$~$ |
---|

"Might one of the following ..." | $~$n$~$ |
---|

"Nice!" | $~$\log_b(x)$~$ |
---|

"Nice!" | $~$b$~$ |
---|

"Nice!" | $~$x$~$ |
---|

"No, the difference between the two sentences li..." | $~$K$~$ |
---|

"No, the difference between the two sentences li..." | $~$O$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$X_i$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$x_i$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$\mathbf {pa}_i$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$x_i$~$ |
---|

"No, this kind of factorization is used for *any..." | $~$\mathbf x$~$ |
---|

"Not 2^100?" | $~$2^{101}$~$ |
---|

"Not clear what this means?" | $~$\prec$~$ |
---|

"Not clear what this means?" | $~$\langle \mathbb R, \leq \rangle$~$ |
---|

"Not clear what this means?" | $~$\leq$~$ |
---|

"Not clear what this means?" | $~$0 < 1$~$ |
---|

"Not clear what this means?" | $~$0$~$ |
---|

"Not clear what this means?" | $~$\mathbb R$~$ |
---|

"Not clear what this means?" | $~$x \in \mathbb R$~$ |
---|

"Not clear what this means?" | $~$x > 0$~$ |
---|

"Not clear what this means?" | $~$y \in \mathbb R$~$ |
---|

"Not clear what this means?" | $~$0 < y < x$~$ |
---|

"Not clear what this means?" | $~$\mathbb R$~$ |
---|

"Okay now I'm also confused...." | $~$f(x)=1$~$ |
---|

"Okay now I'm also confused...." | $~$1$~$ |
---|

"Okay now I'm also confused...." | $~$\{1\}$~$ |
---|

"On "Conditions for Goodhart's curse": It seems ..." | $~$V:s \mapsto V(s)$~$ |
---|

"On "Conditions for Goodhart's curse": It seems ..." | $~$s$~$ |
---|

"On "Conditions for Goodhart's curse": It seems ..." | $~$n$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~${^-2}$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~${^-6}$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~$\log_{10}(10^{-6}) - \log_{10}(10^{-2})$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~${^-4}$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~${^-13.3}$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~$\log_{10}(\frac{0.10}{0.90}) - \log_{10}(\frac{0.11}{0.89}) \approx {^-0.954}-{^-0.907} \approx {^-0.046}$~$ |
---|

"One of these does log( prob/ 1 - prob) the othe..." | $~${^-0.153}$~$ |
---|

"Pedantic remark: Aren't you missing the identit..." | $~$x^{-1}$~$ |
---|

"Pedantic remark: Aren't you missing the identit..." | $~$\rho_{x^{-1}}$~$ |
---|

"Pedantic remark: Aren't you missing the identit..." | $~$\rho_x$~$ |
---|

"Pedantic remark: Aren't you missing the identit..." | $~$\rho_{x^{-1}}$~$ |
---|

"Pedantic remark: Aren't you missing the identit..." | $~$\rho_\epsilon$~$ |
---|

"Seven tenths?" | $~$\log_{10}(500)$~$ |
---|

"Should the p's and q's in o..." | $~$p \prec q$~$ |
---|

"Should the p's and q's in o..." | $~$q$~$ |
---|

"Should the p's and q's in o..." | $~$P$~$ |
---|

"Should the p's and q's in o..." | $~$p$~$ |
---|

"Should the p's and q's in o..." | $~$p \prec q$~$ |
---|

"Should the p's and q's in o..." | $~$p$~$ |
---|

"Should the p's and q's in o..." | $~$q$~$ |
---|

"Should the p's and q's in o..." | $~$p$~$ |
---|

"Should the p's and q's in o..." | $~$q$~$ |
---|

"Should the p's and q's in o..." | $~$q$~$ |
---|

"Should the p's and q's in o..." | $~$p$~$ |
---|

"Smallest?" | $~$x,$~$ |
---|

"Smallest?" | $~$\lceil x \rceil$~$ |
---|

"Smallest?" | $~$\operatorname{ceil}(x),$~$ |
---|

"Smallest?" | $~$n \ge x.$~$ |
---|

"Smallest?" | $~$\lceil 3.72 \rceil = 4, \lceil 4 \rceil = 4,$~$ |
---|

"Smallest?" | $~$\lceil -3.72 \rceil = -3.$~$ |
---|

"Surely they are equivalent. Given a Rice-decidi..." | $~$[n]$~$ |
---|

"Surely they are equivalent. Given a Rice-decidi..." | $~$k$~$ |
---|

"Surely they are equivalent. Given a Rice-decidi..." | $~$[n]$~$ |
---|

"Surely they are equivalent. Given a Rice-decidi..." | $~$k$~$ |
---|

"Thanks for this analysis and congratulations on..." | $~$\pi_5$~$ |
---|

"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|

"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|

"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$(x : y)$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$\alpha$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$(\alpha x : \alpha y).$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$x$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$y$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$\frac{x}{y}.$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$\frac{x}{y}$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$(x : y),$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$\left(\frac{x}{y} : 1\right).$~$ |
---|

"The $x/y$ notation is confusing - these ratios ..." | $~$x/y$~$ |
---|

"The expression P(a_x [ ]-> o_i) is meaningless...." | $~$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$~$ |
---|

"The following would be simpler and more consist..." | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|

"The inverse of multiplication is division. To t..." | $~$1 : 4$~$ |
---|

"The inverse of multiplication is division. To t..." | $~$3 : 1$~$ |
---|

"The inverse of multiplication is division. To t..." | $~$(1 \cdot 3) : (4 \cdot 1) = 3 : 4$~$ |
---|

"The log used to determine number of bits should..." | $~$H$~$ |
---|

"The log used to determine number of bits should..." | $~$\frac{1}{8}$~$ |
---|

"The log used to determine number of bits should..." | $~$\lnot H$~$ |
---|

"The log used to determine number of bits should..." | $~$\frac{1}{4}$~$ |
---|

"The log used to determine number of bits should..." | $~$\lnot H$~$ |
---|

"The log used to determine number of bits should..." | $~$H,$~$ |
---|

"The log used to determine number of bits should..." | $~$\mathbb P(e \mid H)$~$ |
---|

"The log used to determine number of bits should..." | $~$\mathbb P(e \mid \lnot H)$~$ |
---|

"The log used to determine number of bits should..." | $~$\left(\frac{1}{8} : \frac{1}{4}\right)$~$ |
---|

"The log used to determine number of bits should..." | $~$=$~$ |
---|

"The log used to determine number of bits should..." | $~$(1 : 2),$~$ |
---|

"The log used to determine number of bits should..." | $~$H.$~$ |
---|

"The non-existence of a total order on $\mathbb{..." | $~$\mathbb{C}$~$ |
---|

"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|

"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|

"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|

"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|

"The proof of (5) only goes through for $n\in\ma..." | $~$n\in\mathbb{N}$~$ |
---|

"The proof of (5) only goes through for $n\in\ma..." | $~$f(b)=1\Rightarrow f(b^q)=q$~$ |
---|

"The proof of (5) only goes through for $n\in\ma..." | $~$q\in\mathbb{Q}$~$ |
---|

"The proof of (5) only goes through for $n\in\ma..." | $~$f$~$ |
---|

"The urls are displaying as:
https://arbital.com..." | $~$bayes_rule_details,$~$ |
---|

"This "do" notation may seem mysterious, as it i..." | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|

"This confused me at first because I didn't real..." | $~$\mathbb P(X \mid Y)$~$ |
---|

"This confused me at first because I didn't real..." | $~$X$~$ |
---|

"This confused me at first because I didn't real..." | $~$Y$~$ |
---|

"This definition of the real numbers has a bigge..." | $~$\mathbb{N} \setminus \{1, 2, 3, 4, 5\}$~$ |
---|

"This definition of the real numbers has a bigge..." | $~${5}$~$ |
---|

"This definition of the real numbers has a bigge..." | $~$1/8$~$ |
---|

"This does not seem like it'd be transparent, es..." | $~$1$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|

"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$1$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$0$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$1$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|

"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|

"This is slightly confusing,..." | $~$\log_{10}(\text{2,310,426})$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{a}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$a$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$a$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m} \times \frac{1}{n}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{1}{m \times n}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{n}{m} = n \times \frac{1}{m}$~$ |
---|

"This relies on a principle "other way" introduc..." | $~$\frac{n}{m} = n \times \frac{1}{m}$~$ |
---|

"This seems like a straw alt..." | $~$V_i$~$ |
---|

"This seems like a straw alt..." | $~$v_i.$~$ |
---|

"This seems like a straw alt..." | $~$v_i$~$ |
---|

"This seems like a straw alt..." | $~$v_i^*$~$ |
---|

"This seems like a straw alt..." | $~$V_i$~$ |
---|

"This wording suggests the group contains only s..." | $~$X = \{ a, b \}$~$ |
---|

"Underline." | $~$n$~$ |
---|

"Underline." | $~$\sqrt{n}$~$ |
---|

"Underline." | $~$n$~$ |
---|

"Underline." | $~$x$~$ |
---|

"Underline." | $~$x$~$ |
---|

"Underline." | $~$x \cdot x$~$ |
---|

"Underline." | $~$n$~$ |
---|

"Underline." | $~$n$~$ |
---|

"Underline." | $~$\sqrt{n}$~$ |
---|

"Underline." | $~$\log_b(316) \approx \frac{5\log_b(10)}{2}$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3)$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$1$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(6),$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(9)$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^{10}),$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^9)$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^{10}).$~$ |
---|

"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3)$~$ |
---|

"What's $n$ exactly?" | $~$x$~$ |
---|

"What's $n$ exactly?" | $~$x$~$ |
---|

"What's $n$ exactly?" | $~$n$~$ |
---|

"What's $n$ exactly?" | $~$x$~$ |
---|

"What's $n$ exactly?" | $~$n$~$ |
---|

"Where did the '16' come fro..." | $~$(5 : 3 : 2) \cdot (2 : 1 : 5) \cdot (12 : 10 : 1) = (120 : 30 : 10) \cong (12/16 : 3/16 : 1/16)$~$ |
---|

"Why is it called a *decision problem*? As a rea..." | $~$D$~$ |
---|

"Why is it called a *decision problem*? As a rea..." | $~$A$~$ |
---|

"Why is it called a *decision problem*? As a rea..." | $~$A$~$ |
---|

"Why is it called a *decision problem*? As a rea..." | $~$\{0,1\}^*$~$ |
---|

"Would be cool to have an im..." | $~$C_2$~$ |
---|

"Would be cool to have an im..." | $~$2$~$ |
---|

"Would be cool to have an im..." | $~$1$~$ |
---|

"Would be cool to have an im..." | $~$-1$~$ |
---|

"Would be cool to have an im..." | $~$1$~$ |
---|

"Would be cool to have an im..." | $~$-1$~$ |
---|

"Would be cool to have an im..." | $~$f(x)$~$ |
---|

"Would be cool to have an im..." | $~$f(-x)$~$ |
---|

"Would be cool to have an im..." | $~$f(x)$~$ |
---|

"Would be cool to have an im..." | $~$(-1) \times (-1) = 1$~$ |
---|

"Would be cool to have an im..." | $~$f(-(-x)) = f(x)$~$ |
---|

"Would it be appropriate to ..." | $~$P$~$ |
---|

"Would it be appropriate to ..." | $~$\leq$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~${+1}$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~${^+1}$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~$0.01$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~$0.000001$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~$0.11$~$ |
---|

"Wrong, they are exactly the same distances. I r..." | $~$0.100001.$~$ |
---|

"[@2] I think there should b..." | $~$\mathbb P(f\mid e\!=\!\textbf {THT}) = \dfrac{\mathcal L(e\!=\!\textbf{THT}\mid f) \cdot \mathbb P(f)}{\mathbb P(e\!=\!\textbf {THT})} = **\dfrac{(1 - x) \cdot x \cdot (1 - x) \cdot 1}{\int_0^1 (1 - x) \cdot x \cdot (1 - x) \cdot 1 \** \operatorname{d}\!f} = 12 \cdot f(1 - f)^2$~$ |
---|

"[@5hc] Thanks for the edit! I made a couple of ..." | $~$\emptyset$~$ |
---|

"[@5hc]: I've made the appropriate changes to th..." | $~$57$~$ |
---|

"[@5hc]: I've made the appropriate changes to th..." | $~$\mathrm{sin}$~$ |
---|

"in X, **such that**..." | $~$f : X \times X \to X$~$ |
---|

"in X, **such that**..." | $~$x, y, z$~$ |
---|

"in X, **such that**..." | $~$X$~$ |
---|

"in X, **such that**..." | $~$f(x, f(y, z)) = f(f(x, y), z)$~$ |
---|

"in X, **such that**..." | $~$+$~$ |
---|

"in X, **such that**..." | $~$(x + y) + z = x + (y + z)$~$ |
---|

"in X, **such that**..." | $~$x, y,$~$ |
---|

"in X, **such that**..." | $~$z$~$ |
---|

"odd + odd doesn't equal even?" | $~$0 + 2\mathbb Z$~$ |
---|

"odd + odd doesn't equal even?" | $~$1 + 2\mathbb Z$~$ |
---|

"odd + odd doesn't equal even?" | $~$+$~$ |
---|

"odd + odd doesn't equal even?" | $~$\text{even}$~$ |
---|

"odd + odd doesn't equal even?" | $~$\text{odd}$~$ |
---|

"odd + odd doesn't equal even?" | $~$\text{even}+ \text{even} = \text{even}$~$ |
---|

"odd + odd doesn't equal even?" | $~$\text{even} + \text{odd} = \text{odd}$~$ |
---|

"odd + odd doesn't equal even?" | $~$\text{odd} + \text{odd} = \text{odd}$~$ |
---|

"output?" | $~$x$~$ |
---|

"output?" | $~$x$~$ |
---|

"output?" | $~$n$~$ |
---|

"output?" | $~$c$~$ |
---|

"output?" | $~$n$~$ |
---|

"output?" | $~$c.$~$ |
---|

"test" | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$f(x\cdot y)=f(x)+f(y)$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$g$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$g$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$h$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$h(x+y)=h(x)+h(y)$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $$~$h(g(x\cdot y))=h(g(x))+h(g(y))$~$$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$h$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$h(x)=ch(x)$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$c$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{R}$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{Q}$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{R}$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|

"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|

"use colon instead?" | $~$\mathsf{Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {CooperateBot},$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"use colon instead?" | $~$\mathsf {CooperateBot},$~$ |
---|

"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|

"“got” would be clearer." | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1+2+4+8+\dotsc=-1$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$0.999\dots\neq1$~$ |
---|

0.999...=1 | $~$0.999\dots$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$0.999\dots$~$ |
---|

0.999...=1 | $~$9$~$ |
---|

0.999...=1 | $~$\sum_{k=1}^\infty 9 \cdot 10^{-k}$~$ |
---|

0.999...=1 | $~$(\sum_{k=1}^n 9 \cdot 10^{-k})_{n\in\mathbb N}$~$ |
---|

0.999...=1 | $~$a_n$~$ |
---|

0.999...=1 | $~$n$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$\varepsilon>0$~$ |
---|

0.999...=1 | $~$N\in\mathbb N$~$ |
---|

0.999...=1 | $~$n>N$~$ |
---|

0.999...=1 | $~$|1-a_n|<\varepsilon$~$ |
---|

0.999...=1 | $~$1-a_n=10^{-n}$~$ |
---|

0.999...=1 | $~$a_0$~$ |
---|

0.999...=1 | $~$0$~$ |
---|

0.999...=1 | $~$a_0=0$~$ |
---|

0.999...=1 | $~$1-a_0=1=10^0$~$ |
---|

0.999...=1 | $~$1-a_i=10^{-i}$~$ |
---|

0.999...=1 | $~$1-a_n=10^{-n}$~$ |
---|

0.999...=1 | $~$n$~$ |
---|

0.999...=1 | $~$10^{-n}$~$ |
---|

0.999...=1 | $~$10^{-n}$~$ |
---|

0.999...=1 | $~$0.999\dotsc=1$~$ |
---|

0.999...=1 | $~$0.999\dotsc=1$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$0.$~$ |
---|

0.999...=1 | $~$0$~$ |
---|

0.999...=1 | $~$1-0.999\dotsc=0.000\dotsc001\neq0$~$ |
---|

0.999...=1 | $~$0.000\dotsc001$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$0$~$ |
---|

0.999...=1 | $~$0.000\dotsc001$~$ |
---|

0.999...=1 | $~$0$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$0.9, 0.99, 0.999, \dotsc$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$9.999\dotsc$~$ |
---|

0.999...=1 | $~$9$~$ |
---|

0.999...=1 | $~$9.99-0.999=8.991$~$ |
---|

0.999...=1 | $~$9.999\dotsc-0.999\dotsc=8.999\dotsc991$~$ |
---|

0.999...=1 | $~$9$~$ |
---|

0.999...=1 | $~$0.999\dotsc$~$ |
---|

0.999...=1 | $~$8.999\dotsc991$~$ |
---|

0.999...=1 | $~$1$~$ |
---|

A googol | $~$10^{100},$~$ |
---|

A googolplex | $~$10^{10^{100}}$~$ |
---|

A googolplex | $~$10^{googol}$~$ |
---|

A googolplex | $~$ 10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.$~$ |
---|

A quick econ FAQ for AI/ML folks concerned about technological unemployment | $~$1 to be effectively +$~$ |
---|

A quick econ FAQ for AI/ML folks concerned about technological unemployment | $~$E = -mc^2,$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$\theta$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$\theta$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$0$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$1.$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$M$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$N$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $$~$\frac{M + 1}{M + N + 2} : \frac{N + 1}{M + N + 2}$~$$ |
---|

A reply to Francois Chollet on intelligence explosion | $$~$HTHTHTHTHTHTHTHT…$~$$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$H.$~$ |
---|

A reply to Francois Chollet on intelligence explosion | $~$HTTHTTHTTHTT$~$ |
---|

AI control on the cheap | $~$\mathbb{E}$~$ |
---|

AI control on the cheap | $~$\mathbb{E}$~$ |
---|

AI safety mindset | $~$\Sigma_1$~$ |
---|

AI safety mindset | $~$\Sigma_2$~$ |
---|

AIXI | $~$tl$~$ |
---|

AIXI | $~$l$~$ |
---|

AIXI | $~$t$~$ |
---|

AIXI-tl | $~$\text{AIXI}^{tl}$~$ |
---|

AIXI-tl | $~$l$~$ |
---|

AIXI-tl | $~$t$~$ |
---|

AIXI-tl | $~$tl$~$ |
---|

Abelian group | $~$G$~$ |
---|

Abelian group | $~$(X, \bullet)$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$\bullet$~$ |
---|

Abelian group | $~$x, y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x \bullet y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x \bullet y$~$ |
---|

Abelian group | $~$xy$~$ |
---|

Abelian group | $~$x(yz) = (xy)z$~$ |
---|

Abelian group | $~$x, y, z$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$e$~$ |
---|

Abelian group | $~$x$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xe=ex=x$~$ |
---|

Abelian group | $~$x$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x^{-1}$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|

Abelian group | $~$x, y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xy=yx$~$ |
---|

Abelian group | $~$G=(X, \bullet)$~$ |
---|

Abelian group | $~$\bullet$~$ |
---|

Abelian group | $~$x, y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x \bullet y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x \bullet y$~$ |
---|

Abelian group | $~$xy$~$ |
---|

Abelian group | $~$x(yz) = (xy)z$~$ |
---|

Abelian group | $~$x, y, z$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$e$~$ |
---|

Abelian group | $~$x$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xe=ex=x$~$ |
---|

Abelian group | $~$x$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$x^{-1}$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|

Abelian group | $~$x, y$~$ |
---|

Abelian group | $~$X$~$ |
---|

Abelian group | $~$xy=yx$~$ |
---|

Abelian group | $~$\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\}$~$ |
---|

Abelian group | $~$aba^{-1}db^{-1}=d^{-1}$~$ |
---|

Abelian group | $~$aa^{-1}bb^{-1}d=d^{-1}$~$ |
---|

Abelian group | $~$d=d^{-1}$~$ |
---|

Abelian group | $~$aba^{-1}$~$ |
---|

Abelian group | $~$aa^{-1}b$~$ |
---|

Ability to read logic | $~$(\exists v: \forall w > v: \forall x>0, y>0, z>0: x^w + y^w \neq z^w) \rightarrow ((1 = 0) \vee (1 + 0 = 0 + 1))$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$1 - p$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$1 - p$~$ |
---|

Absent-Minded Driver dilemma | $~$p^2$~$ |
---|

Absent-Minded Driver dilemma | $~$0(1-p) + 4(1-p)p + 1p^2$~$ |
---|

Absent-Minded Driver dilemma | $~$4 -6p$~$ |
---|

Absent-Minded Driver dilemma | $~$p = \frac{2}{3}$~$ |
---|

Absent-Minded Driver dilemma | $~$\$0\cdot\frac{1}{3} + \$4\cdot\frac{2}{3}\frac{1}{3} + \$1\cdot\frac{2}{3}\frac{2}{3} = \$\frac{4}{3} \approx \$1.33.$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$q.$~$ |
---|

Absent-Minded Driver dilemma | $~$1 : q,$~$ |
---|

Absent-Minded Driver dilemma | $~$\frac{1}{1+q}$~$ |
---|

Absent-Minded Driver dilemma | $~$\frac{q}{1+q}$~$ |
---|

Absent-Minded Driver dilemma | $~$p,$~$ |
---|

Absent-Minded Driver dilemma | $~$4p(1-p) + 1p^2.$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$4(1-p) + 1p.$~$ |
---|

Absent-Minded Driver dilemma | $~$\frac{1}{1+q}(4p(1-p) + p^2) + \frac{q}{1+q}(4(1-p) + p)$~$ |
---|

Absent-Minded Driver dilemma | $~$\frac{-6p - 3q + 4}{q+1}$~$ |
---|

Absent-Minded Driver dilemma | $~$p=\frac{4-3q}{6}.$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$q,$~$ |
---|

Absent-Minded Driver dilemma | $~$p=q=\frac{4}{9}.$~$ |
---|

Absent-Minded Driver dilemma | $~$\$4\cdot\frac{4}{9}\frac{5}{9} + \$1\cdot\frac{4}{9}\frac{4}{9} \approx \$1.19.$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$q,$~$ |
---|

Absent-Minded Driver dilemma | $~$1 : q \cong \frac{1}{1+q} : \frac{q}{1+q}$~$ |
---|

Absent-Minded Driver dilemma | $~$p,$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$4p(1-q) + 1pq.$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$4(1-p) + 1p.$~$ |
---|

Absent-Minded Driver dilemma | $~$q,$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $$~$\frac{1}{1+q}(4p(1-q) + pq) + \frac{q}{1+q}(4(1-p) + p)$~$$ |
---|

Absent-Minded Driver dilemma | $~$\frac{4 - 6q}{1+q}$~$ |
---|

Absent-Minded Driver dilemma | $~$p.$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$4-6q = 0 \implies q=\frac{2}{3}.$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$q$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absent-Minded Driver dilemma | $~$p$~$ |
---|

Absolute Complement | $~$A^\complement$~$ |
---|

Absolute Complement | $~$A$~$ |
---|

Absolute Complement | $~$A$~$ |
---|

Absolute Complement | $~$U$~$ |
---|

Absolute Complement | $~$A^\complement = U \setminus A$~$ |
---|

Absolute Complement | $~$A^\complement$~$ |
---|

Absolute Complement | $~$U$~$ |
---|

Absolute Complement | $~$A$~$ |
---|

Ackermann function | $~$A \cdot B = \underbrace{A + A + \ldots A}_{B \text{ copies of } A}$~$ |
---|

Ackermann function | $~$A^B = \underbrace{A \times A \times \ldots A}_{B \text{ copies of } A}$~$ |
---|

Ackermann function | $~$A ^ B$~$ |
---|

Ackermann function | $~$A \uparrow B$~$ |
---|

Ackermann function | $~$A \uparrow\uparrow B = \underbrace{A^{A^{\ldots^A}}}_{B \text{ copies of } A}$~$ |
---|

Ackermann function | $~$\uparrow^n$~$ |
---|

Ackermann function | $~$n$~$ |
---|

Ackermann function | $~$A \uparrow^2 B = \underbrace{A \uparrow^1 (A \uparrow^1 (\ldots A))}_{B \text{ copies of } A}$~$ |
---|

Ackermann function | $~$A \uparrow^n B = \underbrace{A \uparrow^{n-1} (A \uparrow^{n-1} (\ldots A))}_{B \text{ copies of } A}$~$ |
---|

Ackermann function | $~$A(n) = n \uparrow^n n$~$ |
---|

Ackermann function | $~$A(6)$~$ |
---|

Ackermann function | $~$A(1)=1$~$ |
---|

Ackermann function | $~$A(2)=4$~$ |
---|

Ackermann function | $~$A(3)$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{\text{number}}$~$ |
---|

Addition of rational numbers (Math 0) | $~$5$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{\text{number}}$~$ |
---|

Addition of rational numbers (Math 0) | $~$a+b$~$ |
---|

Addition of rational numbers (Math 0) | $~$a$~$ |
---|

Addition of rational numbers (Math 0) | $~$b$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{2}{2} + \frac{3}{3} = 2$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{n}{n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$n$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{8}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$5+8=13$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{8}{3} = \frac{13}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3} = \frac{4}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4} = \frac{3}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3} = \frac{4}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} = \frac{20}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{4} = \frac{15}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$5 \times 3 = 15$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{20}{12} + \frac{15}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{35}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers (Math 0) | $~$2 \times 5 = 10$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{10}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{10}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$m$~$ |
---|

Addition of rational numbers (Math 0) | $~$2$~$ |
---|

Addition of rational numbers (Math 0) | $~$n$~$ |
---|

Addition of rational numbers (Math 0) | $~$5$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{n} = \frac{m}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$m$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m} = \frac{n}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Addition of rational numbers (Math 0) | $~$n$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{\text{thing}}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{\text{thing}}$~$ |
---|

Addition of rational numbers (Math 0) | $$~$\frac{1}{m} + \frac{1}{n} = \frac{n}{m \times n} + \frac{m}{m \times n}$~$$ |
---|

Addition of rational numbers (Math 0) | $$~$\frac{a}{m} + \frac{b}{m} = \frac{a+b}{m}$~$$ |
---|

Addition of rational numbers (Math 0) | $~$a$~$ |
---|

Addition of rational numbers (Math 0) | $~$b$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{4} + \frac{5}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$3 \times 4 = 12$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{15}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$5 \times 3$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{5}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{20}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$5 \times 4 = 20$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $~$\frac{15}{12} + \frac{20}{12} = \frac{35}{12}$~$ |
---|

Addition of rational numbers (Math 0) | $$~$\frac{a}{m} + \frac{b}{n} = \frac{a \times n}{m \times n} + \frac{b \times m}{m \times n} = \frac{a \times n + b \times m}{m \times n}$~$$ |
---|

Addition of rational numbers (Math 0) | $~$a \times n + b \times m$~$ |
---|

Addition of rational numbers (Math 0) | $~$a \times n$~$ |
---|

Addition of rational numbers (Math 0) | $~$b \times m$~$ |
---|

Addition of rational numbers (Math 0) | $~$a, b, m, n$~$ |
---|

Addition of rational numbers (Math 0) | $~$m$~$ |
---|

Addition of rational numbers (Math 0) | $~$n$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{5}$~$ |
---|

Addition of rational numbers exercises | $$~$\frac{1}{10} + \frac{1}{5} = \frac{1 \times 5 + 10 \times 1}{10 \times 5} = \frac{5+10}{50} = \frac{15}{50}$~$$ |
---|

Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
---|

Addition of rational numbers exercises | $~$15$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{50}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5} = \frac{2}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{2}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{15} + \frac{1}{10}$~$ |
---|

Addition of rational numbers exercises | $$~$\frac{1}{10} + \frac{1}{15} = \frac{1 \times 15 + 10 \times 1}{10 \times 15} = \frac{25}{150} = \frac{1}{6}$~$$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{30}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{15}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{3}{30} + \frac{2}{30} = \frac{5}{30}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{5}{30} = \frac{1}{6}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{25}{150} = \frac{1}{6}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{15}$~$ |
---|

Addition of rational numbers exercises | $$~$\frac{1}{15} + \frac{1}{10} = \frac{1 \times 10 + 15 \times 1}{15 \times 10} = \frac{25}{150} = \frac{1}{6}$~$$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{15} = \frac{1}{15} + \frac{1}{10}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{6}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{0}{5} + \frac{2}{5}$~$ |
---|

Addition of rational numbers exercises | $~$5$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers exercises | $~$0$~$ |
---|

Addition of rational numbers exercises | $~$2$~$ |
---|

Addition of rational numbers exercises | $~$2$~$ |
---|

Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{0}{7} + \frac{2}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
---|

Addition of rational numbers exercises | $$~$\frac{0}{7} + \frac{2}{5} = \frac{0 \times 5 + 2 \times 7}{5 \times 7} = \frac{0 + 14}{35} = \frac{14}{35}$~$$ |
---|

Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{5} + \frac{-1}{10}$~$ |
---|

Addition of rational numbers exercises | $$~$\frac{1}{15} + \frac{-1}{10} = \frac{1 \times 10 + 15 \times (-1)}{15 \times 10} = \frac{10 - 15}{150} = \frac{-5}{150} = \frac{-1}{30}$~$$ |
---|

Addition of rational numbers exercises | $~$\frac{7}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{13}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{a}{b}$~$ |
---|

Addition of rational numbers exercises | $~$a$~$ |
---|

Addition of rational numbers exercises | $~$b$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
---|

Addition of rational numbers exercises | $~$7$~$ |
---|

Addition of rational numbers exercises | $~$13$~$ |
---|

Addition of rational numbers exercises | $~$6$~$ |
---|

Addition of rational numbers exercises | $~$\frac{6}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{3}{4}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{7}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{13}{7}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{a}{b}$~$ |
---|

Addition of rational numbers exercises | $~$a$~$ |
---|

Addition of rational numbers exercises | $~$b$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{8 \times 7} = \frac{1}{56}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{7 \times 7}{7 \times 8} = \frac{49}{56}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{8 \times 13}{8 \times 7} = \frac{104}{56}$~$ |
---|

Addition of rational numbers exercises | $~$49$~$ |
---|

Addition of rational numbers exercises | $~$104$~$ |
---|

Addition of rational numbers exercises | $~$55$~$ |
---|

Addition of rational numbers exercises | $~$\frac{1}{56}$~$ |
---|

Addition of rational numbers exercises | $~$\frac{55}{56}$~$ |
---|

Advanced agent properties | $~$\mathbb P(Y|X)$~$ |
---|

Advanced agent properties | $~$X$~$ |
---|

Advanced agent properties | $~$Y$~$ |
---|

Advanced agent properties | $~$Y,$~$ |
---|

Advanced agent properties | $~$X.$~$ |
---|

Advanced agent properties | $~$X$~$ |
---|

Advanced agent properties | $~$Y.$~$ |
---|

Advanced nonagent | $~$\pi_0$~$ |
---|

Advanced nonagent | $~$\mathbb E [U | \operatorname{do}(\pi_0), HumansObeyPlan]$~$ |
---|

Advanced nonagent | $~$\mathbb E [U | \operatorname{do}(\pi_0)],$~$ |
---|

Algebraic field | $~$(R, +, \times)$~$ |
---|

Algebraic field | $~$R$~$ |
---|

Algebraic field | $~$1$~$ |
---|

Algebraic field | $~$0$~$ |
---|

Algebraic field | $~$r \in R$~$ |
---|

Algebraic field | $~$x \in R$~$ |
---|

Algebraic field | $~$xr = rx = 1$~$ |
---|

Algebraic field | $~$0 \not = 1$~$ |
---|

Algebraic structure | $~$X$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$a \circ (b * c) = (a \circ b) * (a \circ c)$~$ |
---|

Algebraic structure tree | $~$(a * b) \circ c = (a \circ c) * (b \circ c)$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$a \circ (a * b) = a * (a \circ b) = a$~$ |
---|

Algebraic structure tree | $~$*$~$ |
---|

Algebraic structure tree | $~$\circ$~$ |
---|

Algebraic structure tree | $~$\wedge$~$ |
---|

Algebraic structure tree | $~$\vee$~$ |
---|

Algorithmic complexity | $~$3\uparrow\uparrow\uparrow3$~$ |
---|

All you need for SAT Math Here! | $~$\frac{y_2-y_1}{x_2-x_1}=\frac{rise}{run}=tan\theta$~$ |
---|

All you need for SAT Math Here! | $~$y=mx+b\rightarrow slope=m$~$ |
---|

All you need for SAT Math Here! | $~$ax+by=c\rightarrow slope=\frac{-a}{b}$~$ |
---|

All you need for SAT Math Here! | $~$\rightarrow$~$ |
---|

All you need for SAT Math Here! | $~$\rightarrow$~$ |
---|

All you need for SAT Math Here! | $~$\rightarrow$~$ |
---|

All you need for SAT Math Here! | $~$\rightarrow$~$ |
---|

All you need for SAT Math Here! | $~$y=mx+{b_1}, y=mx+{b_2}, {b_1}\neq {b_2}$~$ |
---|

All you need for SAT Math Here! | $~$y=mx+{b_1}, y=\frac{-1}{m}x+{b_2}$~$ |
---|

All you need for SAT Math Here! | $~${a_1}x+{b_1}y={c_1}$~$ |
---|

All you need for SAT Math Here! | $~${a_2}x+{b_2}y={c_2}$~$ |
---|

All you need for SAT Math Here! | $~$\frac{a_1}{a_2}\neq \frac{b_1}{b_2}$~$ |
---|

All you need for SAT Math Here! | $~$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}$~$ |
---|

All you need for SAT Math Here! | $~$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$~$ |
---|

All you need for SAT Math Here! | $~$\big(x-h)^2+\big(y-k)^2=r^2$~$ |
---|

All you need for SAT Math Here! | $~$\big(h,k)$~$ |
---|

All you need for SAT Math Here! | $~$r=\sqrt{r^2}$~$ |
---|

All you need for SAT Math Here! | $~${x^2}+{y^2}+{ax}+{by}+c=0$~$ |
---|

All you need for SAT Math Here! | $~$\big(\frac{-a}{2},\frac{-b}{2})$~$ |
---|

All you need for SAT Math Here! | $~$\sqrt{\big(\frac{a}{2})^2+(\frac{b}{2})^2-c}$~$ |
---|

All you need for SAT Math Here! | $~$\big({x_1},{y_1})$~$ |
---|

All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2<r^2$~$ |
---|

All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2=r^2$~$ |
---|

All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2>r^2$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$S_n$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$S_n$~$ |
---|

Alternating group | $~$S_n$~$ |
---|

Alternating group | $~$(132)$~$ |
---|

Alternating group | $~$(13)(23)$~$ |
---|

Alternating group | $~$(1354)$~$ |
---|

Alternating group | $~$(54)(34)(14)$~$ |
---|

Alternating group | $~$A_4$~$ |
---|

Alternating group | $~$(12)(34)$~$ |
---|

Alternating group | $~$(13)(24)$~$ |
---|

Alternating group | $~$(14)(23)$~$ |
---|

Alternating group | $~$(123)$~$ |
---|

Alternating group | $~$(124)$~$ |
---|

Alternating group | $~$(134)$~$ |
---|

Alternating group | $~$(234)$~$ |
---|

Alternating group | $~$(132)$~$ |
---|

Alternating group | $~$(143)$~$ |
---|

Alternating group | $~$(142)$~$ |
---|

Alternating group | $~$(243)$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$2$~$ |
---|

Alternating group | $~$S_n$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$S_n$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$3$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group | $~$A_n$~$ |
---|

Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|

Alternating group is generated by its three-cycles | $~$3$~$ |
---|

Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|

Alternating group is generated by its three-cycles | $~$3$~$ |
---|

Alternating group is generated by its three-cycles | $~$3$~$ |
---|

Alternating group is generated by its three-cycles | $~$(ij)(kl) = (ijk)(jkl)$~$ |
---|

Alternating group is generated by its three-cycles | $~$(ij)(jk) = (ijk)$~$ |
---|

Alternating group is generated by its three-cycles | $~$(ij)(ij) = e$~$ |
---|

Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|

Alternating group is generated by its three-cycles | $~$3$~$ |
---|

Alternating group is generated by its three-cycles | $~$3$~$ |
---|

Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|

Alternating group is generated by its three-cycles | $~$(ijk) = (ij)(jk)$~$ |
---|

An early stage prioritisation model | $$~$ \textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills} $~$$ |
---|

An early stage prioritisation model | $$~$ \textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills} $~$$ |
---|

An introductory guide to modern logic | $~$\phi$~$ |
---|

An introductory guide to modern logic | $~$\phi$~$ |
---|

An introductory guide to modern logic | $~$=, \wedge, \implies$~$ |
---|

An introductory guide to modern logic | $~$0$~$ |
---|

An introductory guide to modern logic | $~$n+1$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$\forall n. 0 \not = n+1$~$ |
---|

An introductory guide to modern logic | $~$\forall$~$ |
---|

An introductory guide to modern logic | $~$A\implies B$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$B$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$w$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$w$~$ |
---|

An introductory guide to modern logic | $~$w$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$\phi$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$\phi$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \phi$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$\phi$~$ |
---|

An introductory guide to modern logic | $~$1$~$ |
---|

An introductory guide to modern logic | $~$=$~$ |
---|

An introductory guide to modern logic | $~$1$~$ |
---|

An introductory guide to modern logic | $~$a$~$ |
---|

An introductory guide to modern logic | $~$0$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$2^{a_1}3^{a_2}5^{a_3}\cdots p(n)^{a_n}$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$Axiom(x)$~$ |
---|

An introductory guide to modern logic | $~$IsEqualTo42(x)$~$ |
---|

An introductory guide to modern logic | $~$x = 42$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash IsEqualTo42(42)$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \exists x IsEqualTo42(x)$~$ |
---|

An introductory guide to modern logic | $~$PA\not\vdash IsEqualTo42(7)$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash Axiom(\textbf{n})$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$n+1$~$ |
---|

An introductory guide to modern logic | $~$Rule(p_1, p_2,…, p_n, r)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$p_1, …., p_n$~$ |
---|

An introductory guide to modern logic | $~$r$~$ |
---|

An introductory guide to modern logic | $~$Proof(x,y)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$x$~$ |
---|

An introductory guide to modern logic | $~$y$~$ |
---|

An introductory guide to modern logic | $~$\exists x. Proof(x,y)$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}(y)$~$ |
---|

An introductory guide to modern logic | $~$\exists$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}(x)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$x$~$ |
---|

An introductory guide to modern logic | $~$\ulcorner 1+1=2 \urcorner$~$ |
---|

An introductory guide to modern logic | $~$1+1=2$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner 1+1=2 \urcorner)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$1+1=2$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$Proof(x,y)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash A$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner)$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\rightarrow B\urcorner) \rightarrow [\square_{PA}(\ulcorner A \urcorner)\rightarrow \square_{PA}(\ulcorner B \urcorner)]$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow \square_{PA} \square_{PA} (\ulcorner A\urcorner)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$B$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$B$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$A\rightarrow B$~$ |
---|

An introductory guide to modern logic | $~$B$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}(\ulcorner A \urcorner)$~$ |
---|

An introductory guide to modern logic | $~$\phi(x)$~$ |
---|

An introductory guide to modern logic | $~$\psi$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \psi \leftrightarrow \phi(\ulcorner \psi \urcorner)$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash A$~$ |
---|

An introductory guide to modern logic | $~$PA\not\vdash A$~$ |
---|

An introductory guide to modern logic | $~$PA\not\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$PA\vdash Proof(\textbf n, \ulcorner A\urcorner)$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$Proof(\textbf n,\ulcorner A\urcorner)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$n$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$P\wedge \neg P$~$ |
---|

An introductory guide to modern logic | $~$P\wedge \neg P$~$ |
---|

An introductory guide to modern logic | $~$P$~$ |
---|

An introductory guide to modern logic | $~$\bot$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA\not \vdash \neg \square_{PA}(\bot)$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$\square_{PA}(\ulcorner A\urcorner$~$ |
---|

An introductory guide to modern logic | $~$A$~$ |
---|

An introductory guide to modern logic | $~$PA$~$ |
---|

Antisymmetric relation | $~$R$~$ |
---|

Antisymmetric relation | $~$(aRb ∧ bRa) → a = b$~$ |
---|

Antisymmetric relation | $~$a ≠ b → (¬aRb ∨ ¬bRa)$~$ |
---|

Antisymmetric relation | $~$aRa$~$ |
---|

Antisymmetric relation | $~$\{(0,0), (1,1), (2,2)…\}$~$ |
---|

Antisymmetric relation | $~$\{(0,1), (1,2), (2,3), (3,4)…\}$~$ |
---|

Antisymmetric relation | $~$\{…(9,3),(10,5),(10,2),(14,7),(14,2)…)\}$~$ |
---|

Arbital Markdown | $~$ax^2 + bx + c = 0$~$ |
---|

Arbital Markdown | $~$ax^2 + bx + c = 0$~$ |
---|

Arbital Markdown | $$~$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$~$$ |
---|

Arbital Markdown | $$~$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$~$$ |
---|

Arbital examplar pages | $~$n^\text{th}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$\Delta_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy | $~$\Delta_0$~$ |
---|

Arithmetical hierarchy | $~$\Pi_0$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_0$~$ |
---|

Arithmetical hierarchy | $~$\forall x < 10: \exists y < x: x + y < 10$~$ |
---|

Arithmetical hierarchy | $~$x, y, z…$~$ |
---|

Arithmetical hierarchy | $~$\phi(x, y, z…)$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n,$~$ |
---|

Arithmetical hierarchy | $~$\forall x: \forall y: \forall z: … \phi(x, y, z…)$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$\Delta_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy | $~$\forall x$~$ |
---|

Arithmetical hierarchy | $~$\exists y$~$ |
---|

Arithmetical hierarchy | $~$\phi(x, y) \leftrightarrow [(x + y) = (y + x)],$~$ |
---|

Arithmetical hierarchy | $~$x$~$ |
---|

Arithmetical hierarchy | $~$y$~$ |
---|

Arithmetical hierarchy | $~$\Delta_0 = \Pi_0 = \Sigma_0.$~$ |
---|

Arithmetical hierarchy | $~$+$~$ |
---|

Arithmetical hierarchy | $~$=$~$ |
---|

Arithmetical hierarchy | $~$\Delta_0$~$ |
---|

Arithmetical hierarchy | $~$c$~$ |
---|

Arithmetical hierarchy | $~$d$~$ |
---|

Arithmetical hierarchy | $~$c + d = d + c$~$ |
---|

Arithmetical hierarchy | $~$\forall x_1: \forall x_2: …$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$x_i$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n+1}.$~$ |
---|

Arithmetical hierarchy | $~$\forall x: (x + 3) = (3 + x)$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1.$~$ |
---|

Arithmetical hierarchy | $~$\exists x_1: \exists x_2: …$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_{n+1}.$~$ |
---|

Arithmetical hierarchy | $~$\exists y: \forall x: (x + y) = (y + x)$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_2$~$ |
---|

Arithmetical hierarchy | $~$\exists y: \exists x: (x + y) = (y + x)$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1.$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$\Delta_n.$~$ |
---|

Arithmetical hierarchy | $~$\Delta_0$~$ |
---|

Arithmetical hierarchy | $~$\forall x: \exists y < x: (x + y) = (y + x)$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy | $~$c,$~$ |
---|

Arithmetical hierarchy | $~$\forall x < c: \phi(x)$~$ |
---|

Arithmetical hierarchy | $~$\phi(0) \wedge \phi(1) … \wedge \phi(c)$~$ |
---|

Arithmetical hierarchy | $~$\exists x < c: \phi(x)$~$ |
---|

Arithmetical hierarchy | $~$\phi(0) \vee \phi(1) \vee …$~$ |
---|

Arithmetical hierarchy | $~$z = 2^x \cdot 3^y$~$ |
---|

Arithmetical hierarchy | $~$\Delta_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\exists$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|

Arithmetical hierarchy | $~$\Pi_{n}$~$ |
---|

Arithmetical hierarchy | $~$\forall$~$ |
---|

Arithmetical hierarchy | $~$\phi \in \Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\phi$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy | $~$\Pi_n.$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy | $~$\phi \in \Delta_0$~$ |
---|

Arithmetical hierarchy | $~$\exists x: \phi(x)$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy | $~$\phi$~$ |
---|

Arithmetical hierarchy | $~$\forall x: \phi(x)$~$ |
---|

Arithmetical hierarchy | $~$\phi$~$ |
---|

Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy | $~$\Pi_1.$~$ |
---|

Arithmetical hierarchy | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Delta_0,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_0,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_0$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Delta_0,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$(x + y) > 10^9$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_2,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_{n+1}$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_{n+1}$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_n$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_n$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Delta_n.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Delta_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Sigma_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x = 2,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$2^2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c^c,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$c=1.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$z = 2^x \cdot 3^y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^3 + y^3 = z^3$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$w$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$w = 2^x \cdot 3^y \cdot 5^z$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^3 + y^3 = z^3.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1,$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x^w + y^w = z^w.$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$X \rightarrow Y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$Y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$X$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$X$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$Y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$x$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$y = f(x) = 4x+1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$4x+1$~$ |
---|

Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|

Arity (of a function) | $~$f(a, b, c, d) = ac - bd$~$ |
---|

Arity (of a function) | $~$+$~$ |
---|

Arity (of a function) | $~$(\mathrm{People} \times \mathrm{Ages})$~$ |
---|

Associative operation | $~$\bullet : X \times X \to X$~$ |
---|

Associative operation | $~$x, y, z$~$ |
---|

Associative operation | $~$X$~$ |
---|

Associative operation | $~$x \bullet (y \bullet z) = (x \bullet y) \bullet z$~$ |
---|

Associative operation | $~$+$~$ |
---|

Associative operation | $~$(x + y) + z = x + (y + z)$~$ |
---|

Associative operation | $~$x, y,$~$ |
---|

Associative operation | $~$z$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$x, y,$~$ |
---|

Associative operation | $~$z$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f(f(x, y), z) = f(x, f(y, z)),$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$x$~$ |
---|

Associative operation | $~$y$~$ |
---|

Associative operation | $~$z$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$y$~$ |
---|

Associative operation | $~$z$~$ |
---|

Associative operation | $~$x$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f_3 : X \times X \times X \to X,$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f$~$ |
---|

Associative operation | $~$f_4, f_5, \ldots,$~$ |
---|

Associative operation | $~$\bullet$~$ |
---|

Associative operation | $~$2 \cdot 3 \cdot 4 \cdot 5$~$ |
---|

Associativity vs commutativity | $~$x$~$ |
---|

Associativity vs commutativity | $~$y,$~$ |
---|

Associativity vs commutativity | $~$y$~$ |
---|

Associativity vs commutativity | $~$x.$~$ |
---|

Associativity vs commutativity | $~$a \cdot (b \cdot (c \cdot d)),$~$ |
---|

Associativity vs commutativity | $~$((a \cdot b) \cdot c) \cdot d.$~$ |
---|

Associativity vs commutativity | $~$\cdot$~$ |
---|

Associativity vs commutativity | $~$3 + 2 + (-7) + 5 + (-2) + (-3) + 7,$~$ |
---|

Associativity vs commutativity | $~$3 - 3 + 2 - 2 + 7 - 7 + 5 = 5,$~$ |
---|

Associativity: Examples | $~$(x + y) + z = x + (y + z)$~$ |
---|

Associativity: Examples | $~$x, y,$~$ |
---|

Associativity: Examples | $~$z.$~$ |
---|

Associativity: Examples | $~$n$~$ |
---|

Associativity: Examples | $~$n$~$ |
---|

Associativity: Examples | $~$(x \times y) \times z = x \times (y \times z)$~$ |
---|

Associativity: Examples | $~$x, y,$~$ |
---|

Associativity: Examples | $~$z.$~$ |
---|

Associativity: Examples | $~$n$~$ |
---|

Associativity: Examples | $~$n$~$ |
---|

Associativity: Examples | $~$x$~$ |
---|

Associativity: Examples | $~$y$~$ |
---|

Associativity: Examples | $~$z$~$ |
---|

Associativity: Examples | $~$(x \times y) \times z$~$ |
---|

Associativity: Examples | $~$x \times (y \times z).$~$ |
---|

Associativity: Examples | $~$x$~$ |
---|

Associativity: Examples | $~$y$~$ |
---|

Associativity: Examples | $~$z$~$ |
---|

Associativity: Examples | $~$z$~$ |
---|

Associativity: Examples | $~$(5-3)-2=0$~$ |
---|

Associativity: Examples | $~$5-(3-2)=4.$~$ |
---|

Associativity: Examples | $~$\uparrow$~$ |
---|

Associativity: Examples | $~$\uparrow$~$ |
---|

Associativity: Examples | $~$\uparrow\downarrow.$~$ |
---|

Associativity: Examples | $~$\uparrow\downarrow$~$ |
---|

Associativity: Examples | $~$\uparrow,$~$ |
---|

Associativity: Examples | $~$\uparrow\downarrow\downarrow,$~$ |
---|

Associativity: Examples | $~$\uparrow$~$ |
---|

Associativity: Examples | $~$\uparrow\downarrow,$~$ |
---|

Associativity: Examples | $~$\uparrow\downarrow\uparrow,$~$ |
---|

Associativity: Examples | $~$?$~$ |
---|

Associativity: Examples | $~$(red\ ?\ green)\ ?\ blue = blue$~$ |
---|

Associativity: Examples | $~$red\ ?\ (green\ ?\ blue)=red.$~$ |
---|

Associativity: Intuition | $~$f : X \times X \to X$~$ |
---|

Associativity: Intuition | $~$X$~$ |
---|

Associativity: Intuition | $~$3 + 4 + 5 + 6,$~$ |
---|

Associativity: Intuition | $~$+$~$ |
---|

Associativity: Intuition | $~$[a, b, c, d, \ldots]$~$ |
---|

Associativity: Intuition | $~$a$~$ |
---|

Associativity: Intuition | $~$b$~$ |
---|

Associativity: Intuition | $~$[a, b]$~$ |
---|

Associativity: Intuition | $~$c$~$ |
---|

Associativity: Intuition | $~$b$~$ |
---|

Associativity: Intuition | $~$c$~$ |
---|

Associativity: Intuition | $~$[b, c]$~$ |
---|

Associativity: Intuition | $~$a$~$ |
---|

Associativity: Intuition | $~$[a, b, c]$~$ |
---|

Associativity: Intuition | $~$f : X \times X \to Y$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$+$~$ |
---|

Associativity: Intuition | $~$n$~$ |
---|

Associativity: Intuition | $~$n$~$ |
---|

Associativity: Intuition | $~$+$~$ |
---|

Associativity: Intuition | $~$x$~$ |
---|

Associativity: Intuition | $~$y$~$ |
---|

Associativity: Intuition | $~$z$~$ |
---|

Associativity: Intuition | $~$x$~$ |
---|

Associativity: Intuition | $~$y$~$ |
---|

Associativity: Intuition | $~$z$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f(red,blue)=red,$~$ |
---|

Associativity: Intuition | $~$f(red,green)=green,$~$ |
---|

Associativity: Intuition | $~$f(blue,blue)=blue,$~$ |
---|

Associativity: Intuition | $~$f(blue,green=blue).$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f(f(red, green), blue))=blue,$~$ |
---|

Associativity: Intuition | $~$f(red, f(green, blue))=red.$~$ |
---|

Associativity: Intuition | $~$f(green, blue)$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Associativity: Intuition | $~$f$~$ |
---|

Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0$~$$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$x$~$ |
---|

Asymptotic Notation | $~$f(x) = x$~$ |
---|

Asymptotic Notation | $~$g(x) = x^2$~$ |
---|

Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{x}{x^2} = 0$~$ |
---|

Asymptotic Notation | $~$x = o(x^2)$~$ |
---|

Asymptotic Notation | $~$x^2$~$ |
---|

Asymptotic Notation | $~$x$~$ |
---|

Asymptotic Notation | $~$x$~$ |
---|

Asymptotic Notation | $~$\frac{g(x)}{f(x)}$~$ |
---|

Asymptotic Notation | $~$g(x) - f(x)$~$ |
---|

Asymptotic Notation | $~$x$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) \in o(g(x))$~$ |
---|

Asymptotic Notation | $~$o(g(x))$~$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$f(x) = 200x + 10000$~$ |
---|

Asymptotic Notation | $~$g(x) = x^2$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$x$~$ |
---|

Asymptotic Notation | $~$g(x) > f(x)$~$ |
---|

Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim{x \rightarrow \infty} \frac{200x + 10000}{x^2} = 0$~$$ |
---|

Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{200x + 10000}{x^2} = \lim_{x \rightarrow \infty} \frac{200}{2x}$~$$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = 20x^2 - 10x + 5$~$ |
---|

Asymptotic Notation | $~$g(x) = 2x^2 - x + 10$~$ |
---|

Asymptotic Notation | $~$g(x) = o(f(x))$~$ |
---|

Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = \lim_{x \rightarrow \infty} \frac{2x^2 - x + 10}{20x^2 - 10x + 5} = \lim_{x \rightarrow \infty} \frac{4x - 1}{40x - 10}$~$$ |
---|

Asymptotic Notation | $$~$= \lim_{x \rightarrow \infty} \frac{4}{40} = \frac{1}{10}$~$$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$g(x) \neq o(f(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $$~$\forall_{c>0} \exists_{n>0} \text{ such that } \forall_{x>n} c \cdot f(x) \leq g(x)$~$$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$200 x + 10000 = o(x^2)$~$ |
---|

Asymptotic Notation | $~$c$~$ |
---|

Asymptotic Notation | $~$c(200x + 10000)$~$ |
---|

Asymptotic Notation | $~$x^2$~$ |
---|

Asymptotic Notation | $~$n$~$ |
---|

Asymptotic Notation | $~$f(x) = o(f(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ g(x) \neq o(f(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x)) \text{ and } g(x) = o(h(x))\ \ \implies\ \ f(x)= o(h(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ c + f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ c \cdot f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = 1$~$ |
---|

Asymptotic Notation | $~$f(x) = log(log(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = log(x)$~$ |
---|

Asymptotic Notation | $~$f(x) = x$~$ |
---|

Asymptotic Notation | $~$f(x) = x \cdot log(x)$~$ |
---|

Asymptotic Notation | $~$f(x) = x^{1+\epsilon}$~$ |
---|

Asymptotic Notation | $~$0 < \epsilon < 1$~$ |
---|

Asymptotic Notation | $~$f(x) = x^2$~$ |
---|

Asymptotic Notation | $~$f(x) = x^3$~$ |
---|

Asymptotic Notation | $~$f(x) = x^4$~$ |
---|

Asymptotic Notation | $~$f(x) = e^{cx}$~$ |
---|

Asymptotic Notation | $~$f(x) = x!$~$ |
---|

Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0$~$ |
---|

Asymptotic Notation | $~$0 < \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} < \infty$~$ |
---|

Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|

Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \infty$~$ |
---|

Asymptotic Notation | $~$f(x) = \omega(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$g(x) = \omega(f(x))$~$ |
---|

Asymptotic Notation | $~$g(x)$~$ |
---|

Asymptotic Notation | $~$f(x)$~$ |
---|

Asymptotic Notation | $~$o(g(x))$~$ |
---|

Asymptotic Notation | $~$\Theta(g(x))$~$ |
---|

Asymptotic Notation | $~$\omega(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = O(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = \Omega(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = \omega(g(x))$~$ |
---|

Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|

Asymptotic Notation | $~$\Theta(n\ lg(n))$~$ |
---|

Asymptotic Notation | $~$\Theta(n^2)$~$ |
---|

Asymptotic Notation | $~$n\ lg(n)$~$ |
---|

Asymptotic Notation | $~$n^2$~$ |
---|

Asymptotic Notation | $~$n lg(n) = o(n^2)$~$ |
---|

Asymptotic Notation | $~$[6,5,4,3,2,1]$~$ |
---|

Asymptotic Notation | $~$[1,2,3,4,6,5]$~$ |
---|

Asymptotic Notation | $~$n$~$ |
---|

Asymptotic Notation | $~$n^2$~$ |
---|

Asymptotic Notation | $~$O(n^2)$~$ |
---|

Author's guide to Arbital | $~$e$~$ |
---|

Author's guide to Arbital | $~$\approx 2.718…$~$ |
---|

Axiom | $~$T$~$ |
---|

Axiom | $~$\forall w. weight(w)\rightarrow 0<w \wedge w < 1$~$ |
---|

Axiom | $~$0$~$ |
---|

Axiom | $~$1$~$ |
---|

Axiom | $~$[P(0) \wedge \forall n. P(n)\rightarrow P(n+1)]\rightarrow \forall n. P(n)$~$ |
---|

Axiom | $~$PA$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$Y \in X$~$ |
---|

Axiom of Choice | $~$Y$~$ |
---|

Axiom of Choice | $~$f$~$ |
---|

Axiom of Choice | $~$Y$~$ |
---|

Axiom of Choice | $~$f(Y) \in Y$~$ |
---|

Axiom of Choice | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$Y_1, Y_2, Y_3$~$ |
---|

Axiom of Choice | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|

Axiom of Choice | $~$f$~$ |
---|

Axiom of Choice | $~$f(Y_1) = y_1$~$ |
---|

Axiom of Choice | $~$f(Y_2) = y_2$~$ |
---|

Axiom of Choice | $~$f(Y_3) = y_3$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|

Axiom of Choice | $~$f$~$ |
---|

Axiom of Choice | $~$Y$~$ |
---|

Axiom of Choice | $~$n$~$ |
---|

Axiom of Choice | $~$n$~$ |
---|

Axiom of Choice | $~$f$~$ |
---|

Axiom of Choice | $~$X_1, X_2, X_3, \ldots$~$ |
---|

Axiom of Choice | $~$\prod_{i \in \mathbb{N}} X_i$~$ |
---|

Axiom of Choice | $~$(x_1, x_2, x_3, \ldots )$~$ |
---|

Axiom of Choice | $~$x_1 \in X_1$~$ |
---|

Axiom of Choice | $~$x_2 \in X_2$~$ |
---|

Axiom of Choice | $~$X_1$~$ |
---|

Axiom of Choice | $~$X_2$~$ |
---|

Axiom of Choice | $~$X_3$~$ |
---|

Axiom of Choice | $~$f: C \rightarrow C$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$x_0$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$x_0 \in C$~$ |
---|

Axiom of Choice | $~$f(x_0) = x_0$~$ |
---|

Axiom of Choice | $~$(x , y)$~$ |
---|

Axiom of Choice | $~$x$~$ |
---|

Axiom of Choice | $~$y$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$(A_i)_{i \in I}$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$A_n$~$ |
---|

Axiom of Choice | $~$\mathcal{U}$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$I$~$ |
---|

Axiom of Choice | $~$\mathcal{U}$~$ |
---|

Axiom of Choice | $~$\mathcal{U}$~$ |
---|

Axiom of Choice | $~$X \subseteq I$~$ |
---|

Axiom of Choice | $~$X \in \mathcal{U}$~$ |
---|

Axiom of Choice | $~$(A_i)_{i \in X}$~$ |
---|

Axiom of Choice | $~$(A_i)_{i \in I}$~$ |
---|

Axiom of Choice | $~$A$~$ |
---|

Axiom of Choice | $~$A_i$~$ |
---|

Axiom of Choice | $~$A$~$ |
---|

Axiom of Choice | $~$A_i$~$ |
---|

Axiom of Choice | $~$A_i$~$ |
---|

Axiom of Choice | $~$A_i$~$ |
---|

Axiom of Choice | $~$A$~$ |
---|

Axiom of Choice | $~$A$~$ |
---|

Axiom of Choice | $~$\in$~$ |
---|

Axiom of Choice | $~$x \in X$~$ |
---|

Axiom of Choice | $~$x$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$\in$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$\phi$~$ |
---|

Axiom of Choice | $~$\in$~$ |
---|

Axiom of Choice | $~$\{x \in X : \phi(x) \}$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$\phi$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$x$~$ |
---|

Axiom of Choice | $~$\phi(x)$~$ |
---|

Axiom of Choice | $~$A, B, C, \ldots$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$xy = yx$~$ |
---|

Axiom of Choice | $~$x$~$ |
---|

Axiom of Choice | $~$y$~$ |
---|

Axiom of Choice | $~$xy \not= yx$~$ |
---|

Axiom of Choice | $~$S_3$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$A$~$ |
---|

Axiom of Choice | $~$A \times A$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$Y$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$u \in X$~$ |
---|

Axiom of Choice | $~$C$~$ |
---|

Axiom of Choice | $~$u \geq c$~$ |
---|

Axiom of Choice | $~$c \in C$~$ |
---|

Axiom of Choice | $~$m \in X$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$x \in X$~$ |
---|

Axiom of Choice | $~$m \not< x$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$m$~$ |
---|

Axiom of Choice | $~$V$~$ |
---|

Axiom of Choice | $~$V$~$ |
---|

Axiom of Choice | $~$v_1 \in V$~$ |
---|

Axiom of Choice | $~$v$~$ |
---|

Axiom of Choice | $~$\{v_1\}$~$ |
---|

Axiom of Choice | $~$\{v_1\} \subseteq \{v, v_2\} \subseteq \{v_1, v_2, v_3 \} \subseteq \cdots$~$ |
---|

Axiom of Choice | $~$ \{v_1\} \cup \{v_1, v_2\} \cup \{v_1, v_2, v_3 \} \cdots = \{v_1, v_2, v_3, \ldots \}$~$ |
---|

Axiom of Choice | $~$B$~$ |
---|

Axiom of Choice | $~$B$~$ |
---|

Axiom of Choice | $~$v_i$~$ |
---|

Axiom of Choice | $~$B$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$V$~$ |
---|

Axiom of Choice | $~$V$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$v \in V$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$M \cup \{v\}$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$v$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$M$~$ |
---|

Axiom of Choice | $~$V$~$ |
---|

Axiom of Choice | $~$\{v_1\}$~$ |
---|

Axiom of Choice | $~$\{v_1, v_2\}$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$\{42, 48, 64, \ldots\}$~$ |
---|

Axiom of Choice | $~$42$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$R$~$ |
---|

Axiom of Choice | $~$(x_n)_{n \in \mathbb{N}}$~$ |
---|

Axiom of Choice | $~$x_n$~$ |
---|

Axiom of Choice | $~$R$~$ |
---|

Axiom of Choice | $~$x_{n+1}$~$ |
---|

Axiom of Choice | $~$\mathbb{N}$~$ |
---|

Axiom of Choice | $~$\mathbb{R}$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$X$~$ |
---|

Axiom of Choice | $~$P(X)$~$ |
---|

Axiom of Choice | $~$\mathbb{R}$~$ |
---|

Axiom of Choice | $~$P(\mathbb{N})$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y \in X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f(Y) \in Y$~$ |
---|

Axiom of Choice Definition (Intuitive) | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y_1, Y_2, Y_3$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f(Y_1) = y_1$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f(Y_2) = y_2$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f(Y_3) = y_3$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$n$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$n$~$ |
---|

Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|

Bag | $~$\operatorname{Bag}(1, 1, 2, 3) = \operatorname{Bag}(2, 1, 3, 1) \neq \operatorname{Bag}(1, 2, 3).$~$ |
---|

Bayes' rule | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|

Bayes' rule | $~$h_1$~$ |
---|

Bayes' rule | $~$\mathbb {P}(h_1)$~$ |
---|

Bayes' rule | $~$\mathbb {P}(h_2)$~$ |
---|

Bayes' rule | $~$e_0$~$ |
---|

Bayes' rule | $~$e_0$~$ |
---|

Bayes' rule | $~$h_1$~$ |
---|

Bayes' rule | $~$\mathbb {P}(e_0\mid h_1)$~$ |
---|

Bayes' rule | $~$\mathbb {P}(e_0\mid h_2)$~$ |
---|

Bayes' rule | $~$e_0$~$ |
---|

Bayes' rule | $~$h_2$~$ |
---|

Bayes' rule | $~$e_0$~$ |
---|

Bayes' rule | $~$h_1$~$ |
---|

Bayes' rule | $~$h_2$~$ |
---|

Bayes' rule | $$~$\frac{\mathbb {P}(h_1\mid e_0)}{\mathbb {P}(h_2\mid e_0)} = \frac{\mathbb {P}(h_1)}{\mathbb {P}(h_2)} \cdot \frac{\mathbb {P}(e_0\mid h_1)}{\mathbb {P}(e_0\mid h_2)}$~$$ |
---|

Bayes' rule | $~$\mathbb P(\mathbf{H}\mid e) \propto \operatorname{\mathbb {P}}(e\mid \mathbf{H}) \cdot \operatorname{\mathbb {P}}(\mathbf{H}).$~$ |
---|

Bayes' rule: Definition | $~$H_1$~$ |
---|

Bayes' rule: Definition | $~$H_2$~$ |
---|

Bayes' rule: Definition | $~$e_0.$~$ |
---|

Bayes' rule: Definition | $~$\mathbb P(H_i)$~$ |
---|

Bayes' rule: Definition | $~$H_i$~$ |
---|

Bayes' rule: Definition | $~$\mathbb P(e_0\mid H_i)$~$ |
---|

Bayes' rule: Definition | $~$e_0$~$ |
---|

Bayes' rule: Definition | $~$H_i$~$ |
---|

Bayes' rule: Definition | $~$\mathbb P(H_i\mid e_0)$~$ |
---|

Bayes' rule: Definition | $~$H_i$~$ |
---|

Bayes' rule: Definition | $~$e_0.$~$ |
---|

Bayes' rule: Definition | $$~$\dfrac{\mathbb P(H_1)}{\mathbb P(H_2)} \times \dfrac{\mathbb P(e_0\mid H_1)}{\mathbb P(e_0\mid H_2)} = \dfrac{\mathbb P(H_1\mid e_0)}{\mathbb P(H_2\mid e_0)}$~$$ |
---|

Bayes' rule: Definition | $~$h_i$~$ |
---|

Bayes' rule: Definition | $~$\alpha$~$ |
---|

Bayes' rule: Definition | $~$h_k$~$ |
---|

Bayes' rule: Definition | $~$\beta$~$ |
---|

Bayes' rule: Definition | $~$h_i$~$ |
---|

Bayes' rule: Definition | $~$h_k$~$ |
---|

Bayes' rule: Definition | $~$h_i$~$ |
---|

Bayes' rule: Definition | $~$\alpha \cdot \beta$~$ |
---|

Bayes' rule: Definition | $~$h_k.$~$ |
---|

Bayes' rule: Definition | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|

Bayes' rule: Definition | $~$\mathbb P(X\wedge Y) = \mathbb P(X\mid Y) \cdot \mathbb P(Y):$~$ |
---|

Bayes' rule: Definition | $$~$
\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)}
= \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e \wedge H_j)}
= \dfrac{\mathbb P(e \wedge H_i) / \mathbb P(e)}{\mathbb P(e \wedge H_j) / \mathbb P(e)}
= \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}
$~$$ |
---|

Bayes' rule: Definition | $$~$\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
$~$$ |
---|

Bayes' rule: Definition | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) \cong & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & \cong (8 : 2 : 3)
\end{array}$~$$ |
---|

Bayes' rule: Definition | $~$\mathbb P(H_i\mid e),$~$ |
---|

Bayes' rule: Definition | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|

Bayes' rule: Definition | $$~$\mathbb P(\mathbf{H}\mid e) \propto \mathbb P(e\mid \mathbf{H}) \cdot \mathbb P(\mathbf{H}).$~$$ |
---|

Bayes' rule: Definition | $~$1,$~$ |
---|

Bayes' rule: Functional form | $$~$\mathbb P(H_x\mid e) \propto \mathcal L_e(H_x) \cdot \mathbb P(H_x)$~$$ |
---|

Bayes' rule: Functional form | $$~$\mathbb P(H_x\mid e) \propto \mathcal L_e(H_x) \cdot \mathbb P(H_x)$~$$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb P(b),$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb P(b)\cdot \mathrm{d}b$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$[b + \mathrm{d}b]$~$ |
---|

Bayes' rule: Functional form | $~$\mathrm db$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$[a, b]$~$ |
---|

Bayes' rule: Functional form | $~$\int_a^b \mathbb P(b) \, \mathrm db.$~$ |
---|

Bayes' rule: Functional form | $~$b,$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb P(b) = 1$~$ |
---|

Bayes' rule: Functional form | $~$b,$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb P(b)\, \mathrm db = \mathrm db$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b)$~$ |
---|

Bayes' rule: Functional form | $~$t_1$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b = 0.6,$~$ |
---|

Bayes' rule: Functional form | $~$b = 0.33,$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b)$~$ |
---|

Bayes' rule: Functional form | $~$t_1$~$ |
---|

Bayes' rule: Functional form | $~$b,$~$ |
---|

Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b) = 1 - b.$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb O(b\mid t_1) = \mathcal L_{t_1}(b) \cdot \mathbb P(b) = 1 - b,$~$ |
---|

Bayes' rule: Functional form | $~$\int_0^1 (1 - b) \, \mathrm db = \frac{1}{2}.$~$ |
---|

Bayes' rule: Functional form | $~$\mathbb P(b \mid t_1) = \dfrac{\mathbb O(b \mid t_1)}{\int_0^1 \mathbb O(b \mid t_1) \, \mathrm db} = 2 \cdot (1 - f)$~$ |
---|

Bayes' rule: Functional form | $~$h_2t_3.$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b$~$ |
---|

Bayes' rule: Functional form | $~$b.$~$ |
---|

Bayes' rule: Functional form | $$~$\mathbb P(b \mid t_1h_2t_3) = \frac{\mathcal L_{t_1h_2t_3}(b) \cdot \mathbb P(b)}{\mathbb P(t_1h_2t_3)} = \frac{(1 - b) \cdot b \cdot (1 - b) \cdot 1}{\int_0^1 (1 - b) \cdot b \cdot (1 - b) \cdot 1 \, \mathrm{d}b} = {12\cdot b(1 - b)^2}$~$$ |
---|

Bayes' rule: Log-odds form | $~$H_i$~$ |
---|

Bayes' rule: Log-odds form | $~$H_j$~$ |
---|

Bayes' rule: Log-odds form | $~$e$~$ |
---|

Bayes' rule: Log-odds form | $$~$
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right ).
$~$$ |
---|

Bayes' rule: Log-odds form | $~$H_i$~$ |
---|

Bayes' rule: Log-odds form | $~$H_j$~$ |
---|

Bayes' rule: Log-odds form | $~$e$~$ |
---|

Bayes' rule: Log-odds form | $$~$
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right ).
$~$$ |
---|

Bayes' rule: Log-odds form | $~$(1 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(1 : 2) \times (4 : 1) \times (2 : 1),$~$ |
---|

Bayes' rule: Log-odds form | $~$(1 \times 4 \times 2 : 2 \times 1 \times 1) = (8 : 2) = (4 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$2$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_2 (\frac{1}{1}) = 0$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_2 (\frac{1}{2}) = {-1}$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_2 (\frac{4}{1}) = {+2}$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_2 (\frac{2}{1}) = {+1}$~$ |
---|

Bayes' rule: Log-odds form | $~$0 + {^-1} + {^+2} + {^+1} = {^+2}$~$ |
---|

Bayes' rule: Log-odds form | $~$(2^{+2} : 1) = (4 : 1),$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$\lnot H,$~$ |
---|

Bayes' rule: Log-odds form | $~$2 : 1$~$ |
---|

Bayes' rule: Log-odds form | $~$H.$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$(1 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(2 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(4 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(8 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(16 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$(32 : 1).$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{1}{2} = 50\%$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{2}{3} \approx 67\%$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{4}{5} = 80\%$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{8}{9} \approx 89\%$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{16}{17} \approx 94\%$~$ |
---|

Bayes' rule: Log-odds form | $~$\frac{32}{33} \approx 97\%.$~$ |
---|

Bayes' rule: Log-odds form | $~$(2 : 1)$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$-\infty$~$ |
---|

Bayes' rule: Log-odds form | $~$+\infty$~$ |
---|

Bayes' rule: Log-odds form | $~$(0,1)$~$ |
---|

Bayes' rule: Log-odds form | $~${+1}$~$ |
---|

Bayes' rule: Log-odds form | $~${^+1}$~$ |
---|

Bayes' rule: Log-odds form | $~$0.01$~$ |
---|

Bayes' rule: Log-odds form | $~$0.000001$~$ |
---|

Bayes' rule: Log-odds form | $~$0.11$~$ |
---|

Bayes' rule: Log-odds form | $~$0.100001.$~$ |
---|

Bayes' rule: Log-odds form | $~${^-2}$~$ |
---|

Bayes' rule: Log-odds form | $~${^-6}$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_{10}(10^{-6}) - \log_{10}(10^{-2})$~$ |
---|

Bayes' rule: Log-odds form | $~${^-4}$~$ |
---|

Bayes' rule: Log-odds form | $~${^-13.3}$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_{10}(\frac{0.10}{0.90}) - \log_{10}(\frac{0.11}{0.89}) \approx {^-0.954}-{^-0.907} \approx {^-0.046}$~$ |
---|

Bayes' rule: Log-odds form | $~${^-0.153}$~$ |
---|

Bayes' rule: Log-odds form | $~$2 : 1,$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$H$~$ |
---|

Bayes' rule: Log-odds form | $~$1 : 2$~$ |
---|

Bayes' rule: Log-odds form | $~${^-3}$~$ |
---|

Bayes' rule: Log-odds form | $~${^-1}$~$ |
---|

Bayes' rule: Log-odds form | $~${^-4}$~$ |
---|

Bayes' rule: Log-odds form | $~$(1 : 16)$~$ |
---|

Bayes' rule: Log-odds form | $~$\mathbb P({positive}\mid {HIV}) = .997$~$ |
---|

Bayes' rule: Log-odds form | $~$\mathbb P({negative}\mid \neg {HIV}) = .998$~$ |
---|

Bayes' rule: Log-odds form | $~$\mathbb P({positive} \mid \neg {HIV}) = .002.$~$ |
---|

Bayes' rule: Log-odds form | $~$1 : 100,000$~$ |
---|

Bayes' rule: Log-odds form | $~$500 : 1.$~$ |
---|

Bayes' rule: Log-odds form | $~$\log_{10}(500) \approx 2.7$~$ |
---|

Bayes' rule: Log-odds form | $~$500 : 1$~$ |
---|

Bayes' rule: Log-odds form | $~$0$~$ |
---|

Bayes' rule: Log-odds form | $~$1$~$ |
---|

Bayes' rule: Log-odds form | $~$-\infty$~$ |
---|

Bayes' rule: Log-odds form | $~$+\infty,$~$ |
---|

Bayes' rule: Log-odds form | $~$0$~$ |
---|

Bayes' rule: Log-odds form | $~$1$~$ |
---|

Bayes' rule: Log-odds form | $~$0$~$ |
---|

Bayes' rule: Log-odds form | $~$1$~$ |
---|

Bayes' rule: Log-odds form | $~$\mathbb P(X) + \mathbb P(\lnot X)$~$ |
---|

Bayes' rule: Log-odds form | $~$\lnot X$~$ |
---|

Bayes' rule: Log-odds form | $~$X$~$ |
---|

Bayes' rule: Log-odds form | $~$\aleph_0$~$ |
---|

Bayes' rule: Log-odds form | $~$o$~$ |
---|

Bayes' rule: Log-odds form | $~$e = 10\log_{10}(o)$~$ |
---|

Bayes' rule: Odds form | $~$(1 : 2) \times (10 : 1) = (10 : 2) = (5 : 1)$~$ |
---|

Bayes' rule: Odds form | $~$e,$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H \mid e)$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$e$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$\mathcal L_e(\boldsymbol H).$~$ |
---|

Bayes' rule: Odds form | $~$(1 : 2) \times (10 : 1) = (10 : 2) = (5 : 1)$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H = (H_1, H_2, H_3),$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|

Bayes' rule: Odds form | $~$(6 : 2 : 1),$~$ |
---|

Bayes' rule: Odds form | $~$H_1$~$ |
---|

Bayes' rule: Odds form | $~$H_2$~$ |
---|

Bayes' rule: Odds form | $~$H_3.$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H;$~$ |
---|

Bayes' rule: Odds form | $~$H_i$~$ |
---|

Bayes' rule: Odds form | $$~$\mathbb O(\boldsymbol H) \propto \mathbb P(\boldsymbol H).$~$$ |
---|

Bayes' rule: Odds form | $~$H_1$~$ |
---|

Bayes' rule: Odds form | $~$H_2$~$ |
---|

Bayes' rule: Odds form | $~$H_3$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$(H_1, H_2, H_3).$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H) = (80 : 8 : 4) = (20 : 2 : 1)$~$ |
---|

Bayes' rule: Odds form | $~$e_w$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb P(e_w\mid \boldsymbol H) = (0.6, 0.9, 0.3).$~$ |
---|

Bayes' rule: Odds form | $~$\mathcal L_{e_w}(\boldsymbol H) = P(e_w \mid \boldsymbol H).$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H\mid e)$~$ |
---|

Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|

Bayes' rule: Odds form | $~$e.$~$ |
---|

Bayes' rule: Odds form | $$~$\mathbb O(\boldsymbol H) \times \mathcal L_{e}(\boldsymbol H) = \mathbb O(\boldsymbol H\mid e)$~$$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|

Bayes' rule: Odds form | $~$\mathcal L_{e}(\boldsymbol H)$~$ |
---|

Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H\mid e).$~$ |
---|

Bayes' rule: Odds form | $~$\mathcal L_e(\boldsymbol H) = (0.6, 0.9, 0.3)$~$ |
---|

Bayes' rule: Odds form | $~$(2 : 3 : 1).$~$ |
---|

Bayes' rule: Odds form | $~$(20 : 2 : 1).$~$ |
---|

Bayes' rule: Odds form | $~$(0.6 : 0.9 : 0.3)$~$ |
---|

Bayes' rule: Odds form | $~$(2 : 3 : 1).$~$ |
---|

Bayes' rule: Odds form | $~$e_w$~$ |
---|

Bayes' rule: Odds form | $~$(20 : 2 : 1) \times (2 : 3 : 1) = (40 : 6 : 1).$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(2 : 8) \times (9 : 3) \ = \ (1 : 4) \times (3 : 1) \ = \ (3 : 4),$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(x : y)$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(x : y)$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$\alpha$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(\alpha x : \alpha y).$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(1 : 2 : 1)$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$\frac{1}{4} : \frac{2}{4} : \frac{1}{4}.$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(a : b : c)$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(\frac{a}{a + b + c} : \frac{b}{a + b + c} : \frac{c}{a + b + c}).$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$A, B, C$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$\mathbb P(A), \mathbb P(B), \mathbb P(C)$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$1.$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$\textbf{Prior odds} \times \textbf{Likelihood ratio} = \textbf{Posterior odds}$~$ |
---|

Bayes' rule: Odds form (Intro, Math 1) | $~$(1 : 9 ) \times (3 : 1) \ = \ (3 : 9) \ \cong \ (1 : 3)$~$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(X \mid Y) = \frac{\mathbb P(X \wedge Y)}{\mathbb P (Y)}$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(Y) = \sum_k \mathbb P(Y \wedge X_k)$~$ |
---|

Bayes' rule: Probability form | $~$H$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$e,$~$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$e,$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$H.$~$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$\sum_k (\text {expression containing } k)$~$ |
---|

Bayes' rule: Probability form | $~$k$~$ |
---|

Bayes' rule: Probability form | $~$k$~$ |
---|

Bayes' rule: Probability form | $~$\mathbf H$~$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$\mathbf H$~$ |
---|

Bayes' rule: Probability form | $~$H_1, H_2, H_3$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(H_2 \mid heads).$~$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)}{\sum_k \mathbb P(heads \mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)}{[\mathbb P(heads \mid H_1) \cdot \mathbb P(H_1)] + [\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)] + [\mathbb P(heads \mid H_3) \cdot \mathbb P(H_3)]}$~$$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{0.70 \cdot 0.35 }{[0.50 \cdot 0.40] + [0.70 \cdot 0.35] + [0.20 \cdot 0.25]} = \frac{0.245}{0.20 + 0.245 + 0.05} = 0.\overline{49}$~$$ |
---|

Bayes' rule: Probability form | $~$H$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$e,$~$ |
---|

Bayes' rule: Probability form | $~$H_i$~$ |
---|

Bayes' rule: Probability form | $~$e,$~$ |
---|

Bayes' rule: Probability form | $~$e$~$ |
---|

Bayes' rule: Probability form | $~$H.$~$ |
---|

Bayes' rule: Probability form | $~$H_1,H_2,H_3\ldots$~$ |
---|

Bayes' rule: Probability form | $~$1$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(H_k)$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(H_4)=\frac{1}{5}$~$ |
---|

Bayes' rule: Probability form | $~$E,$~$ |
---|

Bayes' rule: Probability form | $~$e_1, e_2, \ldots.$~$ |
---|

Bayes' rule: Probability form | $~$E = e_j,$~$ |
---|

Bayes' rule: Probability form | $~$e_j.$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3,$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3,$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3.$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3.$~$ |
---|

Bayes' rule: Probability form | $~$e_3,$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$e_3.$~$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(H_4 \mid e_3) = \frac{\mathbb P(e_3 \mid H_4) \cdot \mathbb P(H_4)}{\sum_k \mathbb P(e_3 \mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|

Bayes' rule: Probability form | $~$e_j,$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$e_3.$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$e_5$~$ |
---|

Bayes' rule: Probability form | $~$e_5$~$ |
---|

Bayes' rule: Probability form | $~$e_5$~$ |
---|

Bayes' rule: Probability form | $~$e_5,$~$ |
---|

Bayes' rule: Probability form | $~$e_j$~$ |
---|

Bayes' rule: Probability form | $~$e_3,$~$ |
---|

Bayes' rule: Probability form | $~$e_5.$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$e_5$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$H_4$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$e_3$~$ |
---|

Bayes' rule: Probability form | $~$H_k$~$ |
---|

Bayes' rule: Probability form | $~$e_j$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(e \mid GoodDriver)$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(e \mid BadDriver)$~$ |
---|

Bayes' rule: Probability form | $~$\mathbb P(BadDriver)$~$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(X \mid Y) = \frac{\mathbb P(X \wedge Y)}{\mathbb P (Y)}$~$$ |
---|

Bayes' rule: Probability form | $$~$\mathbb P(Y) = \sum_k \mathbb P(Y \wedge X_k)$~$$ |
---|

Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(H_i \wedge e)}{\mathbb P (e)} \tag{defn. conditional prob.}
$~$$ |
---|

Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(e \wedge H_i)}{\sum_k \mathbb P (e \wedge H_k)} \tag {law of marginal prob.}
$~$$ |
---|

Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(e \mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P (e \mid H_k) \cdot \mathbb P(H_k)} \tag {defn. conditional prob.}
$~$$ |
---|

Bayes' rule: Proportional form | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|

Bayes' rule: Proportional form | $~$H_i$~$ |
---|

Bayes' rule: Proportional form | $~$H_j$~$ |
---|

Bayes' rule: Proportional form | $~$e$~$ |
---|

Bayes' rule: Proportional form | $$~$\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)} = \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}$~$$ |
---|

Bayes' rule: Proportional form | $~$(1 : 4) \times (3 : 1) = (3 : 4).$~$ |
---|

Bayes' rule: Proportional form | $~$(1 : 4) \times (3 : 1) = (3 : 4).$~$ |
---|

Bayes' rule: Proportional form | $~$\frac{1}{4} \times \frac{3}{1} = \frac{3}{4},$~$ |
---|

Bayes' rule: Proportional form | $~$0.25 \times 3 = 0.75.$~$ |
---|

Bayes' rule: Proportional form | $~$(0.25 : 1) \cdot (3 : 1) = (0.75 : 1),$~$ |
---|

Bayes' rule: Vector form | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) = & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & = (8 : 2 : 3)
\end{array}$~$$ |
---|

Bayes' rule: Vector form | $~$\mathbf H$~$ |
---|

Bayes' rule: Vector form | $~$H_1, H_2, \ldots$~$ |
---|

Bayes' rule: Vector form | $~$\mathbf H,$~$ |
---|

Bayes' rule: Vector form | $$~$\mathbb O(\mathbf H) \times \mathcal L_e(\mathbf H) = \mathbb O(\mathbf H \mid e)$~$$ |
---|

Bayes' rule: Vector form | $~$\mathbb O(\mathbf H)$~$ |
---|

Bayes' rule: Vector form | $~$H_i$~$ |
---|

Bayes' rule: Vector form | $~$\mathcal L_e(\mathbf H)$~$ |
---|

Bayes' rule: Vector form | $~$H_i$~$ |
---|

Bayes' rule: Vector form | $~$e,$~$ |
---|

Bayes' rule: Vector form | $~$\mathbb O(\mathbf H \mid e)$~$ |
---|

Bayes' rule: Vector form | $~$H_i.$~$ |
---|

Bayes' rule: Vector form | $$~$\begin{array}{r}
\mathbb O(\mathbf H) \\
\times\ \mathcal L_{e_1}(\mathbf H) \\
\times\ \mathcal L_{e_2}(\mathbf H \wedge e_1) \\
\times\ \mathcal L_{e_3}(\mathbf H \wedge e_1 \wedge e_2) \\
= \mathbb O(\mathbf H \mid e_1 \wedge e_2 \wedge e_3)
\end{array}$~$$ |
---|

Bayes' rule: Vector form | $~$H_{fair},$~$ |
---|

Bayes' rule: Vector form | $~$H_{heads}$~$ |
---|

Bayes' rule: Vector form | $~$H_{tails}$~$ |
---|

Bayes' rule: Vector form | $~$(1/2 : 1/3 : 1/6).$~$ |
---|

Bayes' rule: Vector form | $~$(2 : 3 : 1)$~$ |
---|

Bayes' rule: Vector form | $~$(2 : 1 : 3).$~$ |
---|

Bayes' rule: Vector form | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) = & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & = (8 : 2 : 3) = (8/13 : 2/13 : 3/13)
\end{array}$~$$ |
---|

Bayes' rule: Vector form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i)P(H_i)}{\sum_k \mathbb P(e\mid H_k)P(H_k)}$~$$ |
---|

Bayes' rule: Vector form | $~$(5 : 3 : 2)$~$ |
---|

Bayes' rule: Vector form | $$~$\left(\frac{10}{50} : \frac{3}{30} : \frac{10}{20}\right) = \left(\frac{1}{5} : \frac{1}{10} : \frac{1}{2}\right) = (2 : 1 : 5)$~$$ |
---|

Bayes' rule: Vector form | $$~$\left(\frac{30}{50} : \frac{15}{30} : \frac{1}{20}\right) = \left(\frac{3}{5} : \frac{1}{2} : \frac{1}{20}\right) = (12 : 10 : 1)$~$$ |
---|

Bayes' rule: Vector form | $$~$(5 : 3 : 2) \times (2 : 1 : 5) \times (12 : 10 : 1) = (120 : 30 : 10) = \left(\frac{12}{16} : \frac{3}{16} : \frac{1}{16}\right)$~$$ |
---|

Bayes' rule: Vector form | $$~$\mathbb P({workplace}\mid \neg {romance} \wedge {museum}) \neq \mathbb P({workplace}\mid \neg {romance})$~$$ |
---|

Bayes' rule: Vector form | $~$\mathbb P({museum} \wedge {workplace} \mid \neg {romance})$~$ |
---|

Bayes' rule: Vector form | $~$\mathbb P({museum}\mid \neg {romance}) \cdot \mathbb P({workplace}\mid \neg {romance}).$~$ |
---|

Bayesian view of scientific virtues | $~$Grek$~$ |
---|

Bayesian view of scientific virtues | $~$up, down,$~$ |
---|

Bayesian view of scientific virtues | $~$other.$~$ |
---|

Bayesian view of scientific virtues | $~$Thag$~$ |
---|

Bayesian view of scientific virtues | $~$up, down,$~$ |
---|

Bayesian view of scientific virtues | $~$other$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(\cdot\mid Thag)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) + \mathbb P(down\mid Thag) + \mathbb P(other\mid Thag) = 1.$~$ |
---|

Bayesian view of scientific virtues | $~$1/3$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag), \mathbb P(down\mid Thag),$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(other\mid Thag)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek)!$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(up\mid Grek)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(other\mid Grek)$~$ |
---|

Bayesian view of scientific virtues | $~$up,$~$ |
---|

Bayesian view of scientific virtues | $~$up$~$ |
---|

Bayesian view of scientific virtues | $~$other,$~$ |
---|

Bayesian view of scientific virtues | $~$down$~$ |
---|

Bayesian view of scientific virtues | $~$down$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) = 1.$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) = 1$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag) = 1$~$ |
---|

Bayesian view of scientific virtues | $~$1$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0.95$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0.95$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag) = 0.95$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(blue\mid Thag) = 0.90$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(blue\mid \neg Thag) < 0.01$~$ |
---|

Bayesian view of scientific virtues | $~$\dfrac{\mathbb P(Thag\mid blue)}{\mathbb P(\neg Thag\mid blue)} > 90 \cdot \dfrac{\mathbb P(Thag)}{\mathbb P(\neg Thag)}$~$ |
---|

Bayesian view of scientific virtues | $~$H \rightarrow E,$~$ |
---|

Bayesian view of scientific virtues | $~$\neg E$~$ |
---|

Bayesian view of scientific virtues | $~$\neg H$~$ |
---|

Bayesian view of scientific virtues | $~$E,$~$ |
---|

Bayesian view of scientific virtues | $~$H.$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid currentNewton)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid newModel)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid Neptune \wedge Newton),$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid Neptune \wedge Other).$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Einstein)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Newton),$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Other)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(newObservation\mid Other),$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Newton)$~$ |
---|

Bayesian view of scientific virtues | $~$\mathbb P(observation\mid hypothesis)$~$ |
---|

Bayesian view of scientific virtues | $~$observation$~$ |
---|

Bayesian view of scientific virtues | $~$\neg observation$~$ |
---|

Belief revision as probability elimination | $~$\mathbb P$~$ |
---|

Belief revision as probability elimination | $~$\mathbb P$~$ |
---|

Belief revision as probability elimination | $$~$\begin{array}{l|r|r}
& Sick & Healthy \\
\hline
Test + & 18\% & 24\% \\
\hline
Test - & 2\% & 56\%
\end{array}$~$$ |
---|

Binary function | $~$f$~$ |
---|

Binary function | $~$+,$~$ |
---|

Binary function | $~$-,$~$ |
---|

Binary function | $~$\times,$~$ |
---|

Binary function | $~$\div$~$ |
---|

Binary notation | $~$8207$~$ |
---|

Binary notation | $~$(7 \times 10^0) + (0 \times 10^1) + (2 \times 10^2) + (8 \times 10^3)$~$ |
---|

Binary notation | $~$0$~$ |
---|

Binary notation | $~$1$~$ |
---|

Binary notation | $~$11010$~$ |
---|

Binary notation | $~$(0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4)$~$ |
---|

Binary notation | $~$26$~$ |
---|

Bit | $~$\log_2$~$ |
---|

Bit | $~$\mathbb B$~$ |
---|

Bit | $~$2 : 1$~$ |
---|

Bit | $~$\mathbb B$~$ |
---|

Bit | $~$2 : 1$~$ |
---|

Bit | $~$\log_2$~$ |
---|

Bit | $~$\log_2$~$ |
---|

Bit | $~$\log_2$~$ |
---|

Bit | $~$\log_2$~$ |
---|

Bit (abstract) | $~$\mathbb B$~$ |
---|

Bit (abstract) | $~$\mathbb B$~$ |
---|

Bit (abstract) | $~$\mathbb N$~$ |
---|

Bit (abstract) | $~$\mathbb N$~$ |
---|

Bit (abstract) | $~$\mathbb B$~$ |
---|

Bit (abstract) | $~$\mathbb B$~$ |
---|

Bit (of data) | $~$n$~$ |
---|

Bit (of data) | $~$\log_2(n)$~$ |
---|

Bit (of data) | $~$n$~$ |
---|

Bit (of data) | $~$\log_2(n)$~$ |
---|

Bit (of data) | $~$\log_2(10) \approx 3.32$~$ |
---|

Bit (of data) | $~$2^{10}=1024.$~$ |
---|

Bit (of data) | $~$2^{20}=1048576.$~$ |
---|

Bit (of data) | $~$n$~$ |
---|

Bit (of data) | $~$n$~$ |
---|

Bit (of data) | $~$\log_2(n)$~$ |
---|

Boolean | $~$\land$~$ |
---|

Boolean | $~$\lor$~$ |
---|

Boolean | $~$\neg$~$ |
---|

Boolean | $~$\rightarrow$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$ax+by = c$~$ |
---|

Bézout's theorem | $~$x$~$ |
---|

Bézout's theorem | $~$y$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$ax+by = c$~$ |
---|

Bézout's theorem | $~$x$~$ |
---|

Bézout's theorem | $~$y$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$ax+by=c$~$ |
---|

Bézout's theorem | $~$ax+by=c$~$ |
---|

Bézout's theorem | $~$x$~$ |
---|

Bézout's theorem | $~$y$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$ax$~$ |
---|

Bézout's theorem | $~$by$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$c$~$ |
---|

Bézout's theorem | $~$\mathrm{hcf}(a,b) \mid c$~$ |
---|

Bézout's theorem | $~$d$~$ |
---|

Bézout's theorem | $~$d \times \mathrm{hcf}(a,b) = c$~$ |
---|

Bézout's theorem | $~$a, b$~$ |
---|

Bézout's theorem | $~$x$~$ |
---|

Bézout's theorem | $~$y$~$ |
---|

Bézout's theorem | $~$ax + by = \mathrm{hcf}(a,b)$~$ |
---|

Bézout's theorem | $~$a (xd) + b (yd) = d \mathrm{hcf}(a, b) = c$~$ |
---|

Bézout's theorem | $~$d \times \mathrm{hcf}(a,b) = c$~$ |
---|

Bézout's theorem | $~$ax+by$~$ |
---|

Bézout's theorem | $~$a$~$ |
---|

Bézout's theorem | $~$b$~$ |
---|

Bézout's theorem | $~$\mathrm{hcf}(a,b)$~$ |
---|

Bézout's theorem | $~$ax+by=c$~$ |
---|

Bézout's theorem | $~$d$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$1 < 2$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$2<1$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a < b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b < a$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f: A \to B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g: B \to A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h: A \to B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f(a) = b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f^{-1}(b)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f(a) = b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f^{-1}(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f^{-1}(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $$~$\dots, f^{-1}(g^{-1}(a)), g^{-1}(a), a, f(a), g(f(a)), \dots$~$$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g^{-1}(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$gfgf(a) = a$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $$~$\dots g^{-1} f^{-1}(b), f^{-1}(b), b, g(b), f(g(b)), \dots$~$$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g^{-1} f^{-1}(b)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h(a) = g^{-1}(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g(b)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$X$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f: X \to X$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$x$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f(x) = x$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f: A \to B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g: B \to A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P \cup Q$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$R \cup S$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$R$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$S$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$Q$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$A \to B$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$g^{-1}$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$Q$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P \mapsto A \setminus g(B \setminus f(P))$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|

Cantor-Schröder-Bernstein theorem | $~$P = A \setminus g(B \setminus f(P))$~$ |
---|

Cardinality | $~$A$~$ |
---|

Cardinality | $~$A$~$ |
---|

Cardinality | $~$|A|$~$ |
---|

Cardinality | $~$A$~$ |
---|

Cardinality | $~$|A| = n$~$ |
---|

Cardinality | $~$A$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$\{0, …, (n-1)\}$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$\mathbb N$~$ |
---|

Cardinality | $~$\mathbb N$~$ |
---|

Cardinality | $~$|X|$~$ |
---|

Cardinality | $~$X$~$ |
---|

Cardinality | $~$X.$~$ |
---|

Cardinality | $~$X = \{a, b, c, d\}, |X|=4.$~$ |
---|

Cardinality | $~$S$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$S$~$ |
---|

Cardinality | $~$1$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$\{9, 15, 12, 20\}$~$ |
---|

Cardinality | $~$\{1, 2, 3, 4\}$~$ |
---|

Cardinality | $~$m$~$ |
---|

Cardinality | $~$m$~$ |
---|

Cardinality | $~$4$~$ |
---|

Cardinality | $~$S$~$ |
---|

Cardinality | $~$T$~$ |
---|

Cardinality | $~$f : S \to \{1, 2, 3, \ldots, n\}$~$ |
---|

Cardinality | $~$g : \{1, 2, 3, \ldots, n\} \to T$~$ |
---|

Cardinality | $~$g \circ f$~$ |
---|

Cardinality | $~$S$~$ |
---|

Cardinality | $~$T$~$ |
---|

Cardinality | $~$n$~$ |
---|

Cardinality | $~$\aleph_0$~$ |
---|

Cardinality | $~$\aleph_1, \aleph_2, \aleph_3,$~$ |
---|

Cartesian product | $~$A$~$ |
---|

Cartesian product | $~$B,$~$ |
---|

Cartesian product | $~$A \times B,$~$ |
---|

Cartesian product | $~$(a, b)$~$ |
---|

Cartesian product | $~$a \in A$~$ |
---|

Cartesian product | $~$b \in B.$~$ |
---|

Cartesian product | $~$\mathbb B \times \mathbb N$~$ |
---|

Cartesian product | $~$\mathbb B^3 = \mathbb B \times \mathbb B \times \mathbb B$~$ |
---|

Cartesian product | $~$\times$~$ |
---|

Cartesian product | $~$n$~$ |
---|

Cartesian product | $~$n$~$ |
---|

Category (mathematics) | $~$f$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$Y$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$Y$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$Y$~$ |
---|

Category (mathematics) | $~$f$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$f$~$ |
---|

Category (mathematics) | $~$Y$~$ |
---|

Category (mathematics) | $~$f$~$ |
---|

Category (mathematics) | $~$f$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$Y$~$ |
---|

Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|

Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|

Category (mathematics) | $~$g: Y \rightarrow Z$~$ |
---|

Category (mathematics) | $~$X \rightarrow Z$~$ |
---|

Category (mathematics) | $~$g \circ f$~$ |
---|

Category (mathematics) | $~$gf$~$ |
---|

Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|

Category (mathematics) | $~$g: Y \rightarrow Z$~$ |
---|

Category (mathematics) | $~$h:Z \rightarrow W$~$ |
---|

Category (mathematics) | $~$h(gf) = (hg)f$~$ |
---|

Category (mathematics) | $~$X$~$ |
---|

Category (mathematics) | $~$1_X : X \rightarrow X$~$ |
---|

Category (mathematics) | $~$f:W \rightarrow X$~$ |
---|

Category (mathematics) | $~$g:X \rightarrow Y$~$ |
---|

Category (mathematics) | $~$1_X f = f$~$ |
---|

Category (mathematics) | $~$g 1_X = g$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$\text{dom}(f)$~$ |
---|

Category theory | $~$\text{cod}(f)$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$\text{dom}(f) = X$~$ |
---|

Category theory | $~$\text{cod}(f) = Y$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$X \rightarrow Z$~$ |
---|

Category theory | $~$g \circ f$~$ |
---|

Category theory | $~$gf$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$h:Z \rightarrow W$~$ |
---|

Category theory | $~$h(gf) = (hg)f$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$1_X : X \rightarrow X$~$ |
---|

Category theory | $~$f:W \rightarrow X$~$ |
---|

Category theory | $~$g:X \rightarrow Y$~$ |
---|

Category theory | $~$1_X f = f$~$ |
---|

Category theory | $~$g 1_X = g$~$ |
---|

Category theory | $~$(P, \leq)$~$ |
---|

Category theory | $~$x \rightarrow y$~$ |
---|

Category theory | $~$x$~$ |
---|

Category theory | $~$y$~$ |
---|

Category theory | $~$x \leq y$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$X \rightarrow Z$~$ |
---|

Category theory | $~$g \circ f$~$ |
---|

Category theory | $~$gf$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$h:Z \rightarrow W$~$ |
---|

Category theory | $~$h(gf) = (hg)f$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$1_X : X \rightarrow X$~$ |
---|

Category theory | $~$f:W \rightarrow X$~$ |
---|

Category theory | $~$g:X \rightarrow Y$~$ |
---|

Category theory | $~$1_X f = f$~$ |
---|

Category theory | $~$g 1_X = g$~$ |
---|

Category theory | $~$x \in X$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$g$~$ |
---|

Category theory | $~$g(f(x))$~$ |
---|

Category theory | $~$(g \circ f)(x)$~$ |
---|

Category theory | $~$\mathbb{A}, \mathbb{B}, \mathbb{C}$~$ |
---|

Category theory | $~$A, B, C, W, X, Y, Z$~$ |
---|

Category theory | $~$e, f, g, h, u, v, w$~$ |
---|

Category theory | $~$a, b, c, x, y, z$~$ |
---|

Category theory | $~$E, F, G, H$~$ |
---|

Category theory | $~$\alpha, \beta, \gamma, \delta$~$ |
---|

Category theory | $~$\kappa$~$ |
---|

Category theory | $~$\lambda$~$ |
---|

Category theory | $~$\mathbb{C}$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$\mathbb{C}$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$\mathbb{C}$~$ |
---|

Category theory | $~$f: X \rightarrow T$~$ |
---|

Category theory | $~$f: X \rightarrow T$~$ |
---|

Category theory | $~$g: X \rightarrow T$~$ |
---|

Category theory | $~$f=g$~$ |
---|

Category theory | $~$\{a\}$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$f: X \rightarrow \{a\}$~$ |
---|

Category theory | $~$x$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$a$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$P$~$ |
---|

Category theory | $~$f: P \rightarrow X$~$ |
---|

Category theory | $~$g: P \rightarrow Y$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$W$~$ |
---|

Category theory | $~$u: W \rightarrow X$~$ |
---|

Category theory | $~$v:W \rightarrow Y$~$ |
---|

Category theory | $~$h: W \rightarrow P$~$ |
---|

Category theory | $~$fh = u$~$ |
---|

Category theory | $~$gh = v$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$f: X \rightarrow T$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$f: X \leftarrow T$~$ |
---|

Category theory | $~$T'$~$ |
---|

Category theory | $~$T'$~$ |
---|

Category theory | $~$T$~$ |
---|

Category theory | $~$f: T \rightarrow T'$~$ |
---|

Category theory | $~$g: T' \rightarrow T$~$ |
---|

Category theory | $~$gf = 1_T$~$ |
---|

Category theory | $~$fg = 1_{T'}$~$ |
---|

Category theory | $~$f: T \leftarrow T'$~$ |
---|

Category theory | $~$g: T' \leftarrow T$~$ |
---|

Category theory | $~$fg = 1_T$~$ |
---|

Category theory | $~$gf = 1_{T'}$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$g$~$ |
---|

Category theory | $~$\mathbb{A}$~$ |
---|

Category theory | $~$\mathbb{B}$~$ |
---|

Category theory | $~$F$~$ |
---|

Category theory | $~$\mathbb{A}$~$ |
---|

Category theory | $~$\mathbb{B}$~$ |
---|

Category theory | $~$F: \mathbb{A} \rightarrow \mathbb{B}$~$ |
---|

Category theory | $~$F_0:$~$ |
---|

Category theory | $~$\mathbb{A}$~$ |
---|

Category theory | $~$\rightarrow$~$ |
---|

Category theory | $~$\mathbb{B}$~$ |
---|

Category theory | $~$F_1:$~$ |
---|

Category theory | $~$\mathbb{A}$~$ |
---|

Category theory | $~$\rightarrow$~$ |
---|

Category theory | $~$\mathbb{B}$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$F_1(f): F_0(X) \rightarrow F_1(Y)$~$ |
---|

Category theory | $~$F_1(f)$~$ |
---|

Category theory | $~$F_0$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$F_1(f)$~$ |
---|

Category theory | $~$F_0$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$1_X: X \rightarrow X$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$F_1(1_X): F_0(X) \rightarrow F_0(X)$~$ |
---|

Category theory | $~$F_0(X)$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow Z$~$ |
---|

Category theory | $~$F_1(g) \circ F_1(f): F_0(X) \rightarrow F_0(Z)$~$ |
---|

Category theory | $~$F_1(g \circ f): F_0(X) \rightarrow F_0(Z)$~$ |
---|

Category theory | $~$F_0$~$ |
---|

Category theory | $~$F_1$~$ |
---|

Category theory | $~$F$~$ |
---|

Category theory | $~$F(f): F(X) \rightarrow F(Y)$~$ |
---|

Category theory | $~$f: X \rightarrow Y$~$ |
---|

Category theory | $~$g: Y \rightarrow X$~$ |
---|

Category theory | $~$gf = 1_X$~$ |
---|

Category theory | $~$fg = 1_Y$~$ |
---|

Category theory | $~$W$~$ |
---|

Category theory | $~$g,h: W \rightarrow X$~$ |
---|

Category theory | $~$fg = fh$~$ |
---|

Category theory | $~$g = h$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$X$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$Z$~$ |
---|

Category theory | $~$g,h: X \rightarrow Z$~$ |
---|

Category theory | $~$gf = hf$~$ |
---|

Category theory | $~$g = h$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$Y$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$X = Y$~$ |
---|

Category theory | $~$f: X \rightarrow X$~$ |
---|

Category theory | $~$f$~$ |
---|

Category theory | $~$g: Y \rightarrow X$~$ |
---|

Category theory | $~$gf = 1_X$~$ |
---|

Category theory | $~$g: Y \rightarrow X$~$ |
---|

Category theory | $~$fg = 1_Y$~$ |
---|

Cauchy sequence | $~$X$~$ |
---|

Cauchy sequence | $~$d$~$ |
---|

Cauchy sequence | $~$(x_n)_{n=0}^\infty$~$ |
---|

Cauchy sequence | $~$\varepsilon > 0$~$ |
---|

Cauchy sequence | $~$N$~$ |
---|

Cauchy sequence | $~$m, n > N$~$ |
---|

Cauchy sequence | $~$d(x_m, x_n) < \varepsilon$~$ |
---|

Cauchy sequence | $~$|x_m - x_n|$~$ |
---|

Cauchy's theorem on subgroup existence | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|G|$~$ |
---|

Cauchy's theorem on subgroup existence | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $$~$X = \{ (x_1, x_2, \dots, x_p) : x_1 x_2 \dots x_p = e \}$~$$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$(e, e, \dots, e)$~$ |
---|

Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $$~$(h, (x_1, \dots, x_p)) \mapsto (x_2, x_3, \dots, x_p, x_1)$~$$ |
---|

Cauchy's theorem on subgroup existence | $~$h$~$ |
---|

Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$h^i$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$(x_1, \dots, x_p)$~$ |
---|

Cauchy's theorem on subgroup existence | $~$(x_{i+1}, x_{i+2} , \dots, x_p, x_1, \dots, x_i)$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$x_1 x_2 \dots x_p = e$~$ |
---|

Cauchy's theorem on subgroup existence | $$~$x_{i+1} x_{i+2} \dots x_p x_1 \dots x_i = (x_1 \dots x_i)^{-1} (x_1 \dots x_p) (x_1 \dots x_i) = (x_1 \dots x_i)^{-1} e (x_1 \dots x_i) = e$~$$ |
---|

Cauchy's theorem on subgroup existence | $~$0$~$ |
---|

Cauchy's theorem on subgroup existence | $~$(h^i h^j)(x_1, x_2, \dots, x_p) = h^i(h^j(x_1, x_2, \dots, x_p))$~$ |
---|

Cauchy's theorem on subgroup existence | $~$h^{i+j}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$i+j$~$ |
---|

Cauchy's theorem on subgroup existence | $~$j$~$ |
---|

Cauchy's theorem on subgroup existence | $~$i$~$ |
---|

Cauchy's theorem on subgroup existence | $~$i+j$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x} = (x_1, \dots, x_p) \in X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\mathrm{Orb}_{C_p}(\bar{x})$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|C_p| = p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x} \in X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$x_p = (x_1 \dots x_{p-1})^{-1}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|G|$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|G|^{p-1} = |X|$~$ |
---|

Cauchy's theorem on subgroup existence | $~$|\mathrm{Orb}_{C_p}((e, e, \dots, e))| = 1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p-1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p-1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$p \mid |X|$~$ |
---|

Cauchy's theorem on subgroup existence | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\{ \bar{x} \}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x} = (x_1, \dots, x_p)$~$ |
---|

Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|

Cauchy's theorem on subgroup existence | $~$x_i$~$ |
---|

Cauchy's theorem on subgroup existence | $~$(x, x, \dots, x) \in X$~$ |
---|

Cauchy's theorem on subgroup existence | $~$x^p = e$~$ |
---|

Cauchy's theorem on subgroup existence | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x \not = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x^p = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x^i$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$i < p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p=5$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$\{ a, b, c, d, e\}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(e, e, a, b, a)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(e,a,b,a,e)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x \not = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(x, x, \dots, x)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(x, x, \dots, x)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(e, e, a, b, a)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$eeaba$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$aba = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a,b,c,b,b)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$x^p = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$abcbb = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p=5$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a, a, b, e, \cdot)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$b^{-1} a^{-2}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$aabe(a^{-1} a^{-2}) = e$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|X|$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(e,e,\dots,e)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a_1, a_2, \dots, a_p)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a_2, a_3, \dots, a_p, a_1)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a, a, \dots, a)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$A$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $$~$(a_1, a_2, \dots, a_p), (a_2, a_3, \dots, a_p, a_1), \dots, (a_{p-1}, a_p, a_1, \dots, a_{p-2}), (a_p, a_1, a_2, \dots, a_{p-1})$~$$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p=8$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(1,1,2,2,1,1,2,2)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$n=1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$n=p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(e,e,\dots,e)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1} - 1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p=2$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1} - 1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$(a,a,\dots,a)$~$ |
---|

Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|

Causal decision theories | $~$\mathcal U$~$ |
---|

Causal decision theories | $~$\mathcal O$~$ |
---|

Causal decision theories | $~$a_x$~$ |
---|

Causal decision theories | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(a_x \ \square \!\! \rightarrow o_i)$~$$ |
---|

Causal decision theories | $~$operatorname{do}()$~$ |
---|

Causal decision theories | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|

Causal decision theories | $~$a_0$~$ |
---|

Causal decision theories | $~$o_i$~$ |
---|

Causal decision theories | $~$\mathbb P(o_i|a_0).$~$ |
---|

Causal decision theories | $~$a_0,$~$ |
---|

Causal decision theories | $~$a_0.$~$ |
---|

Causal decision theories | $~$O$~$ |
---|

Causal decision theories | $~$\neg O$~$ |
---|

Causal decision theories | $~$O$~$ |
---|

Causal decision theories | $~$K$~$ |
---|

Causal decision theories | $~$O$~$ |
---|

Causal decision theories | $~$\mathbb P(K|\neg O),$~$ |
---|

Causal decision theories | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K).$~$ |
---|

Causal decision theories | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K),$~$ |
---|

Causal decision theories | $~$\mathbb P(K|\neg O).$~$ |
---|

Causal decision theories | $~$\mathbb P(\bullet \ || \ \bullet)$~$ |
---|

Causal decision theories | $~$X_1$~$ |
---|

Causal decision theories | $~$X_2$~$ |
---|

Causal decision theories | $~$X_3$~$ |
---|

Causal decision theories | $~$X_4$~$ |
---|

Causal decision theories | $~$X_5$~$ |
---|

Causal decision theories | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|

Causal decision theories | $~$X_i$~$ |
---|

Causal decision theories | $~$x_i$~$ |
---|

Causal decision theories | $~$\mathbf {pa}_i$~$ |
---|

Causal decision theories | $~$x_i$~$ |
---|

Causal decision theories | $~$\mathbf x$~$ |
---|

Causal decision theories | $$~$\mathbb P(\mathbf x) = \prod_i \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|

Causal decision theories | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|

Causal decision theories | $$~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j)) = \prod_{i \neq j} \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|

Causal decision theories | $~$\mathbf x$~$ |
---|

Causal decision theories | $~$x_j$~$ |
---|

Causal decision theories | $~$\operatorname{do}$~$ |
---|

Causal decision theories | $~$X_j$~$ |
---|

Causal decision theories | $~$0$~$ |
---|

Causal decision theories | $~$\operatorname{do}(X_j=x_j)$~$ |
---|

Causal decision theories | $~$X_j$~$ |
---|

Causal decision theories | $~$\mathbf{pa}_j,$~$ |
---|

Causal decision theories | $~$X_j = x_j$~$ |
---|

Causal decision theories | $~$\operatorname{do}(X_j=x_j)$~$ |
---|

Causal decision theories | $~$X_k$~$ |
---|

Causal decision theories | $~$X_j$~$ |
---|

Causal decision theories | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|

Causal decision theories | $~$\operatorname{do}()$~$ |
---|

Causal decision theories | $~$W, X, Y, Z$~$ |
---|

Causal decision theories | $$~$\begin{array}{r|c|c}
& \text{One-boxing predicted} & \text{Two-boxing predicted} \\
\hline
\text{W: Take both boxes, no fee:} & \$500,500 & \$500 \\ \hline
\text{X: Take only Box B, no fee:} & \$500,000 & \$0 \\ \hline
\text{Y: Take both boxes, pay fee:} & \$1,000,100 & \$100 \\ \hline
\text{Z: Take only Box B, pay fee:} & \$999,100 & -\$900
\end{array}$~$$ |
---|

Causal decision theories | $~$\operatorname{do}()$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G \times G \to G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$(g, h) \mapsto gh$~$ |
---|

Cayley's Theorem on symmetric groups | $~$\Phi: G \to \mathrm{Sym}(G)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$g \mapsto (h \mapsto gh)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$g \in \mathrm{ker}(\Phi)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$\Phi$~$ |
---|

Cayley's Theorem on symmetric groups | $~$(h \mapsto gh)$~$ |
---|

Cayley's Theorem on symmetric groups | $~$gh = h$~$ |
---|

Cayley's Theorem on symmetric groups | $~$h$~$ |
---|

Cayley's Theorem on symmetric groups | $~$g$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$G$~$ |
---|

Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|

Ceiling | $~$x,$~$ |
---|

Ceiling | $~$\lceil x \rceil$~$ |
---|

Ceiling | $~$\operatorname{ceil}(x),$~$ |
---|

Ceiling | $~$n \ge x.$~$ |
---|

Ceiling | $~$\lceil 3.72 \rceil = 4, \lceil 4 \rceil = 4,$~$ |
---|

Ceiling | $~$\lceil -3.72 \rceil = -3.$~$ |
---|

Ceiling | $~$\mathbb R \to \mathbb Z.$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$0,1,2,\dots$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$\lambda x.M$~$ |
---|

Church encoding | $~$M$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$M$~$ |
---|

Church encoding | $~$x\ (x\ (x\ x))$~$ |
---|

Church encoding | $~$((x\ x)\ x)\ x$~$ |
---|

Church encoding | $~$0$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$3$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$\lambda f.\lambda x.M$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$0$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$0=\lambda f.\lambda x.x$~$ |
---|

Church encoding | $~$1$~$ |
---|

Church encoding | $~$1=\lambda f.\lambda x.f\ x$~$ |
---|

Church encoding | $~$2=\lambda f.\lambda x.f\ (f\ x)$~$ |
---|

Church encoding | $~$3=\lambda f.\lambda x.f\ (f\ (f\ x))$~$ |
---|

Church encoding | $~$4=\lambda f.\lambda x.f\ (f\ (f\ (f\ x)))$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$n=\lambda f.\lambda x.f^n(x)$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$S(n)=n+1$~$ |
---|

Church encoding | $~$S$~$ |
---|

Church encoding | $~$\lambda n$~$ |
---|

Church encoding | $~$\lambda f.\lambda x$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$n+1$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$n+1$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $$~$S=\lambda n.\lambda f.\lambda x.f\ (n\ f\ x).$~$$ |
---|

Church encoding | $~$(n\ f\ x)$~$ |
---|

Church encoding | $~$f^n(x)$~$ |
---|

Church encoding | $~$f\ (n\ f\ x)$~$ |
---|

Church encoding | $~$f(f^n(x))=f^{n+1}(x)$~$ |
---|

Church encoding | $~$f\ x$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $$~$S^\prime=\lambda n.\lambda y.\lambda x.n\ f\ (f\ x).$~$$ |
---|

Church encoding | $~$S$~$ |
---|

Church encoding | $~$S^\prime$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$\lambda a.\lambda b.a$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$S$~$ |
---|

Church encoding | $~$S\ 3=4$~$ |
---|

Church encoding | $~$1$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$m+n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$m+n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $$~$+=\lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x)$~$$ |
---|

Church encoding | $~$n\ f\ x$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$m\ f$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$2+3=5$~$ |
---|

Church encoding | $~$2+3$~$ |
---|

Church encoding | $~$+\ 2\ 3$~$ |
---|

Church encoding | $~$\lambda$~$ |
---|

Church encoding | $~$m+n$~$ |
---|

Church encoding | $~$+\ m\ n$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$x$~$ |
---|

Church encoding | $~$m\times n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$m\times n$~$ |
---|

Church encoding | $~$(f^n)^m(x)=f^{m\times n}(x)$~$ |
---|

Church encoding | $~$f$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$\lambda x.n\ f\ x$~$ |
---|

Church encoding | $~$\eta$~$ |
---|

Church encoding | $~$n\ f$~$ |
---|

Church encoding | $~$n\ f$~$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $$~$\times=\lambda m.\lambda n.\lambda f.\lambda x.m\ (n\ f) x$~$$ |
---|

Church encoding | $~$\eta$~$ |
---|

Church encoding | $$~$\times=\lambda m.\lambda n.\lambda f.m\ (n\ f).$~$$ |
---|

Church encoding | $~$m$~$ |
---|

Church encoding | $~$n$~$ |
---|

Church encoding | $~$\times\ 2\ 3=6$~$ |
---|

Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f$~$ |
---|

Church-Turing thesis: Evidence for the Church-Turing thesis | $~$x$~$ |
---|

Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f(x)$~$ |
---|

Church-Turing thesis: Evidence for the Church-Turing thesis | $~$1/2$~$ |
---|

Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f$~$ |
---|

Closure | $~$S$~$ |
---|

Closure | $~$f$~$ |
---|

Closure | $~$f$~$ |
---|

Closure | $~$S$~$ |
---|

Closure | $~$S$~$ |
---|

Closure | $~$f$~$ |
---|

Closure | $~$S$~$ |
---|

Closure | $~$f$~$ |
---|

Closure | $~$x, y, z \in S$~$ |
---|

Closure | $~$f(x, y, z) \in S$~$ |
---|

Closure | $~$\mathbb Z$~$ |
---|

Closure | $~$\mathbb Z_5 = \{0, 1, 2, 3, 4, 5\}$~$ |
---|

Closure | $~$1 + 5$~$ |
---|

Closure | $~$\mathbb Z_5$~$ |
---|

Codomain (of a function) | $~$\operatorname{cod}(f)$~$ |
---|

Codomain (of a function) | $~$f : X \to Y$~$ |
---|

Codomain (of a function) | $~$Y$~$ |
---|

Codomain (of a function) | $~$+$~$ |
---|

Codomain (of a function) | $~$Y$~$ |
---|

Codomain (of a function) | $~$f$~$ |
---|

Codomain (of a function) | $~$Y$~$ |
---|

Codomain (of a function) | $~$\operatorname{square} : \mathbb R \to \mathbb R$~$ |
---|

Codomain (of a function) | $~$+$~$ |
---|

Codomain (of a function) | $~$\mathbb N$~$ |
---|

Codomain (of a function) | $~$\mathbb Z$~$ |
---|

Codomain vs image | $~$X$~$ |
---|

Codomain vs image | $~$Y$~$ |
---|

Codomain vs image | $~$Y$~$ |
---|

Codomain vs image | $~$f : X \to Y$~$ |
---|

Codomain vs image | $~$X$~$ |
---|

Codomain vs image | $~$Y$~$ |
---|

Codomain vs image | $~$Y$~$ |
---|

Codomain vs image | $~$\mathbb R$~$ |
---|

Codomain vs image | $~$f$~$ |
---|

Codomain vs image | $~$X$~$ |
---|

Codomain vs image | $~$I$~$ |
---|

Codomain vs image | $~$Y$~$ |
---|

Codomain vs image | $~$I$~$ |
---|

Codomain vs image | $~$\mathbb N$~$ |
---|

Codomain vs image | $~$2^{65536} − 3$~$ |
---|

Codomain vs image | $~$\{0, 1\},$~$ |
---|

Codomain vs image | $~$\{0, 1\}$~$ |
---|

Codomain vs image | $~$\{0, 1\}$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(X),$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(\neg X),$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(X) + \mathbb P(\neg X) = 1.$~$ |
---|

Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|

Coherent decisions imply consistent utilities | $~$X >_P Y$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\text{onions} >_P \text{pineapple} >_P \text{mushrooms} >_P \text{onions}$~$$ |
---|

Coherent decisions imply consistent utilities | $~$>$~$ |
---|

Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|

Coherent decisions imply consistent utilities | $~$x > y, y > z \implies x > z$~$ |
---|

Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|

Coherent decisions imply consistent utilities | $~$x, y, z$~$ |
---|

Coherent decisions imply consistent utilities | $~$x > y > z > x.$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$0.01$~$ |
---|

Coherent decisions imply consistent utilities | $~$\text{mushroom} >_P \text{pineapple} >_P \text{onion}$~$ |
---|

Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|

Coherent decisions imply consistent utilities | $~$\text{onions} >_P \text{pineapple}.$~$ |
---|

Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|

Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\mathbb P(heads) \cdot U(\text{1 orange}) + \mathbb P(tails) \cdot U(\text{3 plums}) \\
= 0.50 \cdot €2 + 0.50 \cdot €1.5 = €1.75$~$$ |
---|

Coherent decisions imply consistent utilities | $~$1 \cdot U(\text{1 apple}) = €1.$~$ |
---|

Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|

Coherent decisions imply consistent utilities | $~$-0.2$~$ |
---|

Coherent decisions imply consistent utilities | $~$3$~$ |
---|

Coherent decisions imply consistent utilities | $~$0$~$ |
---|

Coherent decisions imply consistent utilities | $~$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$-0.3$~$ |
---|

Coherent decisions imply consistent utilities | $~$27.$~$ |
---|

Coherent decisions imply consistent utilities | $~$0.6$~$ |
---|

Coherent decisions imply consistent utilities | $~$0.7$~$ |
---|

Coherent decisions imply consistent utilities | $~$1.3$~$ |
---|

Coherent decisions imply consistent utilities | $~$1!$~$ |
---|

Coherent decisions imply consistent utilities | $~$1,$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\mathbb P(\text{heads}) \cdot U(\text{0.8 apples}) + \mathbb P(\text{tails}) \cdot U(\text{0.8 apples}) \\
= 0.6 \cdot €0.8 + 0.7 \cdot €0.8 = €1.04.$~$$ |
---|

Coherent decisions imply consistent utilities | $~$X.$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$x$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$x$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$x.$~$ |
---|

Coherent decisions imply consistent utilities | $~$N \cdot \$x$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$N$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$Y$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$Y$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$Y$~$ |
---|

Coherent decisions imply consistent utilities | $~$x$~$ |
---|

Coherent decisions imply consistent utilities | $~$y$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1.$~$ |
---|

Coherent decisions imply consistent utilities | $~$x + y < \$1,$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$Y$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$x + y.$~$ |
---|

Coherent decisions imply consistent utilities | $~$x + y > \$1,$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$x + y.$~$ |
---|

Coherent decisions imply consistent utilities | $~$x + y - \$1 > \$0.$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$X$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$x$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$y,$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$z$~$ |
---|

Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\mathbb P(Q \wedge R) = \mathbb P(Q) \cdot \mathbb P(R \mid Q)$~$$ |
---|

Coherent decisions imply consistent utilities | $~$z = x \cdot y.$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(Q)$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(R \mid Q)$~$ |
---|

Coherent decisions imply consistent utilities | $~$\mathbb P(Q \wedge R),$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$Q$~$ |
---|

Coherent decisions imply consistent utilities | $~$R$~$ |
---|

Coherent decisions imply consistent utilities | $~$A, B, C$~$ |
---|

Coherent decisions imply consistent utilities | $~$X, Y, Z$~$ |
---|

Coherent decisions imply consistent utilities | $~$x, y, z$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\begin{array}{rrrl}
-Ax & + 0 & - Cz & \geqq 0 \\
A(1-x) & - By & - Cz & \geqq 0 \\
A(1-x) & + B(1-y) & + C(1-z) & \geqq 0
\end{array}$~$$ |
---|

Coherent decisions imply consistent utilities | $~$x, y, z \in (0..1)$~$ |
---|

Coherent decisions imply consistent utilities | $~$z = x * y.$~$ |
---|

Coherent decisions imply consistent utilities | $$~$\begin{array}{rcl}
U(\text{gain \$1 million}) & > & 0.9 \cdot U(\text{gain \$5 million}) + 0.1 \cdot U(\text{gain \$0}) \\
0.5 \cdot U(\text{gain \$0}) + 0.5 \cdot U(\text{gain \$1 million}) & > & 0.45 \cdot U(\text{gain \$5 million}) + 0.55 \cdot U(\text{gain \$0})
\end{array}$~$$ |
---|

Coherent decisions imply consistent utilities | $~$L$~$ |
---|

Coherent decisions imply consistent utilities | $~$M$~$ |
---|

Coherent decisions imply consistent utilities | $~$L > M$~$ |
---|

Coherent decisions imply consistent utilities | $~$p > 0$~$ |
---|

Coherent decisions imply consistent utilities | $~$N$~$ |
---|

Coherent decisions imply consistent utilities | $~$p \cdot L + (1-p)\cdot N > p \cdot M + (1-p) \cdot N.$~$ |
---|

Coherent decisions imply consistent utilities | $~$N,$~$ |
---|

Coherent decisions imply consistent utilities | $~$L$~$ |
---|

Coherent decisions imply consistent utilities | $~$M,$~$ |
---|

Coherent decisions imply consistent utilities | $~$L$~$ |
---|

Coherent decisions imply consistent utilities | $~$M$~$ |
---|

Coherent decisions imply consistent utilities | $~$L$~$ |
---|

Coherent decisions imply consistent utilities | $~$M$~$ |
---|

Coherent decisions imply consistent utilities | $~$L$~$ |
---|

Coherent decisions imply consistent utilities | $~$M,$~$ |
---|

Colon-to notation | $~$f : X \to Y$~$ |
---|

Colon-to notation | $~$\to$~$ |
---|

Colon-to notation | $~$f$~$ |
---|

Colon-to notation | $~$X$~$ |
---|

Colon-to notation | $~$Y$~$ |
---|

Colon-to notation | $~$f$~$ |
---|

Colon-to notation | $~$X$~$ |
---|

Colon-to notation | $~$Y$~$ |
---|

Colon-to notation | $~$f$~$ |
---|

Colon-to notation | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|

Colon-to notation | $~$f$~$ |
---|

Colon-to notation | $~$x \mapsto x^2$~$ |
---|

Colon-to notation | $~$f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$~$ |
---|

Colon-to notation | $~$f$~$ |
---|

Colon-to notation | $~$\times$~$ |
---|

Combining vectors | $~$\mathbf u$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf w$~$ |
---|

Combining vectors | $~$\mathbf s$~$ |
---|

Combining vectors | $~$\mathbf u$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf w$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf d$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf d$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf z$~$ |
---|

Combining vectors | $~$\mathbf r$~$ |
---|

Combining vectors | $~$\mathbf s$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$\mathbf v = 3\mathbf {x} + 4 \mathbf {y}$~$ |
---|

Combining vectors | $~$v$~$ |
---|

Combining vectors | $~$v$~$ |
---|

Combining vectors | $~$\mathbf x$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$3$~$ |
---|

Combining vectors | $~$\mathbf y$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$-1$~$ |
---|

Combining vectors | $~$\mathbf v$~$ |
---|

Combining vectors | $~$(3,4)$~$ |
---|

Combining vectors | $~$O$~$ |
---|

Combining vectors | $~$p = O + 2\mathbf x + 3\mathbf y$~$ |
---|

Combining vectors | $~$q = O - 3\mathbf x + \mathbf y$~$ |
---|

Combining vectors | $~$p = (2, 3)$~$ |
---|

Combining vectors | $~$q = (-3,1)$~$ |
---|

Combining vectors | $~$\mathbf s, \mathbf t$~$ |
---|

Combining vectors | $~$p$~$ |
---|

Combining vectors | $~$(2,\frac{1}{2})$~$ |
---|

Combining vectors | $~$q = (-3,2)$~$ |
---|

Communication: magician example | $~$\log_2(2 \times 6 \times 6) \approx 6.17$~$ |
---|

Communication: magician example | $~$A♠$~$ |
---|

Communication: magician example | $~$K♡.$~$ |
---|

Communication: magician example | $~$2 \cdot 6 \cdot 6 = 72$~$ |
---|

Commutative operation | $~$f$~$ |
---|

Commutative operation | $~$X$~$ |
---|

Commutative operation | $~$+$~$ |
---|

Commutative operation | $~$3 + 4 = 4 + 3.$~$ |
---|

Commutativity: Examples | $~$x+y = y+x$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y,$~$ |
---|

Commutativity: Examples | $~$x \times y = y \times x$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y,$~$ |
---|

Commutativity: Examples | $~$x \times y$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$x.$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$x \times y$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$r$~$ |
---|

Commutativity: Examples | $~$p$~$ |
---|

Commutativity: Examples | $~$s$~$ |
---|

Commutativity: Examples | $~$?$~$ |
---|

Commutativity: Examples | $~$r ? p = p,$~$ |
---|

Commutativity: Examples | $~$r ? s = r,$~$ |
---|

Commutativity: Examples | $~$p ? s = s,$~$ |
---|

Commutativity: Examples | $~$r?p=p?r$~$ |
---|

Commutativity: Examples | $~$(r?p)?s=s$~$ |
---|

Commutativity: Examples | $~$r?(p?s)=r.$~$ |
---|

Commutativity: Examples | $~$x / y$~$ |
---|

Commutativity: Examples | $~$y / x$~$ |
---|

Commutativity: Examples | $~$x$~$ |
---|

Commutativity: Examples | $~$y$~$ |
---|

Commutativity: Examples | $~$2 \times 3$~$ |
---|

Commutativity: Examples | $~$3 \times 5$~$ |
---|

Commutativity: Examples | $~$2 \times 3$~$ |
---|

Commutativity: Intuition | $~$f(x, y)$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$f(x, y)$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$x$~$ |
---|

Commutativity: Intuition | $~$y$~$ |
---|

Commutativity: Intuition | $~$\{b, d, e, l, u, r\}$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$X;$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$(x_1, x_2).$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$|X|$~$ |
---|

Commutativity: Intuition | $~$|X|$~$ |
---|

Commutativity: Intuition | $~$f : X^2 \to Y$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$f(x_1, x_2)$~$ |
---|

Commutativity: Intuition | $~$(x_1, x_2);$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap} : X^2 \to X^2$~$ |
---|

Commutativity: Intuition | $~$(x_1, x_2)$~$ |
---|

Commutativity: Intuition | $~$(x_2, x_1),$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap}(X^2)$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap}$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap}(X^2).$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap}(X^2),$~$ |
---|

Commutativity: Intuition | $~$f$~$ |
---|

Commutativity: Intuition | $~$X^2$~$ |
---|

Commutativity: Intuition | $~$\operatorname{swap}$~$ |
---|

Commutativity: Intuition | $~$f(x_1, x_2)=f(x_2, x_1)$~$ |
---|

Commutativity: Intuition | $~$(x_1, x_2)$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$\bigvee \emptyset$~$ |
---|

Complete lattice | $~$\bigvee L$~$ |
---|

Complete lattice | $~$\bigvee \emptyset$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$\bigvee L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$P$~$ |
---|

Complete lattice | $~$A \subseteq P$~$ |
---|

Complete lattice | $~$A^L$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$\{ p \in P \mid \forall a \in A. p \leq a \}$~$ |
---|

Complete lattice | $~$P$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$P$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$a \in A$~$ |
---|

Complete lattice | $~$A^L$~$ |
---|

Complete lattice | $~$a$~$ |
---|

Complete lattice | $~$A^L$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A^L$~$ |
---|

Complete lattice | $~$\bigvee A^L \leq a$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$p \in P$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$p \in A^L$~$ |
---|

Complete lattice | $~$\bigvee A^L$~$ |
---|

Complete lattice | $~$A^L$~$ |
---|

Complete lattice | $~$p \leq \bigvee A^L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$\bigvee \emptyset$~$ |
---|

Complete lattice | $~$\bigvee L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$X$~$ |
---|

Complete lattice | $~$\langle \mathcal P(X), \subseteq \rangle$~$ |
---|

Complete lattice | $~$X$~$ |
---|

Complete lattice | $~$Y \subset \mathcal P(X)$~$ |
---|

Complete lattice | $~$\bigvee Y = \bigcup Y$~$ |
---|

Complete lattice | $~$\bigvee Y = \bigcup Y$~$ |
---|

Complete lattice | $~$A \in Y$~$ |
---|

Complete lattice | $~$A \subseteq \bigcup Y$~$ |
---|

Complete lattice | $~$\bigcup Y$~$ |
---|

Complete lattice | $~$Y$~$ |
---|

Complete lattice | $~$B \in \mathcal P(X)$~$ |
---|

Complete lattice | $~$Y$~$ |
---|

Complete lattice | $~$A \in Y$~$ |
---|

Complete lattice | $~$A \subseteq B$~$ |
---|

Complete lattice | $~$x \in \bigcup Y$~$ |
---|

Complete lattice | $~$x \in A$~$ |
---|

Complete lattice | $~$A \in Y$~$ |
---|

Complete lattice | $~$A \subseteq B$~$ |
---|

Complete lattice | $~$x \in B$~$ |
---|

Complete lattice | $~$\bigcup Y \subseteq B$~$ |
---|

Complete lattice | $~$\bigcup Y$~$ |
---|

Complete lattice | $~$Y$~$ |
---|

Complete lattice | $~$X$~$ |
---|

Complete lattice | $~$F : X \to X$~$ |
---|

Complete lattice | $~$x \in X$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$x \leq F(x)$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F(x) \leq x$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$X$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$A \subseteq X$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$A$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$\nu F$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$F : L \to L$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$\nu F$~$ |
---|

Complete lattice | $~$L = \langle \mathbb R, \leq \rangle$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F(x) = x$~$ |
---|

Complete lattice | $~$x \leq y \implies F(x) = x \leq y = F(y)$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$\mathbb R$~$ |
---|

Complete lattice | $~$\mathbb R$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$\nu F$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$F : L \to L$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$\nu F$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$F : L \to L$~$ |
---|

Complete lattice | $~$L$~$ |
---|

Complete lattice | $~$\mu F$~$ |
---|

Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|

Complete lattice | $~$\nu F$~$ |
---|

Complete lattice | $~$\bigvee \{x \in L \mid x \leq F(x) \}$~$ |
---|

Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|

Complete lattice | $~$\bigvee \{x \in L \mid F(x) \leq x \}$~$ |
---|

Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$U = \{x \in L \mid F(x) \leq x\}$~$ |
---|

Complete lattice | $~$y = \bigwedge U$~$ |
---|

Complete lattice | $~$F(y) = y$~$ |
---|

Complete lattice | $~$V$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$V \subseteq U$~$ |
---|

Complete lattice | $~$y \leq u$~$ |
---|

Complete lattice | $~$u \in U$~$ |
---|

Complete lattice | $~$y \leq v$~$ |
---|

Complete lattice | $~$v \in V$~$ |
---|

Complete lattice | $~$y$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$u \in U$~$ |
---|

Complete lattice | $~$y \leq u$~$ |
---|

Complete lattice | $~$F(y) \leq F(u) \leq u$~$ |
---|

Complete lattice | $~$F(y)$~$ |
---|

Complete lattice | $~$U$~$ |
---|

Complete lattice | $~$y$~$ |
---|

Complete lattice | $~$F(y) \leq y$~$ |
---|

Complete lattice | $~$y \in U$~$ |
---|

Complete lattice | $~$F$~$ |
---|

Complete lattice | $~$F(y) \leq y$~$ |
---|

Complete lattice | $~$F(F(y)) \leq F(y)$~$ |
---|

Complete lattice | $~$F(y) \in U$~$ |
---|

Complete lattice | $~$y$~$ |
---|

Complete lattice | $~$y \leq F(y)$~$ |
---|

Complete lattice | $~$y \leq F(y)$~$ |
---|

Complete lattice | $~$F(y) \leq y$~$ |
---|

Complete lattice | $~$F(y) = y$~$ |
---|

Complex number | $~$z = a + b\textrm{i}$~$ |
---|

Complex number | $~$\textrm{i}$~$ |
---|

Complex number | $~$\textrm{i}=\sqrt{-1}$~$ |
---|

Complex number | $~$5-3$~$ |
---|

Complex number | $~$0$~$ |
---|

Complex number | $~$\frac{1}{2}, \frac{5}{3}$~$ |
---|

Complex number | $~$-\frac{6}{7}$~$ |
---|

Complex number | $~$\sqrt{9}=3$~$ |
---|

Complex number | $~$\sqrt{2}$~$ |
---|

Complex number | $~$\sqrt{}$~$ |
---|

Complex number | $~$\textrm{i}$~$ |
---|

Complex number | $~$\textrm{i}$~$ |
---|

Complex number | $~$x^2+1=0$~$ |
---|

Complex number | $~$\textrm{i}$~$ |
---|

Complex number | $~$\sqrt{-1}$~$ |
---|

Complex number | $~$\textrm{i}$~$ |
---|

Complex number | $~$-a$~$ |
---|

Complex number | $~$\sqrt{-a}=\textrm{i}\sqrt{a}$~$ |
---|

Complexity theory | $~$P$~$ |
---|

Complexity theory | $~$NP$~$ |
---|

Complexity theory | $~$221$~$ |
---|

Complexity theory | $~$13$~$ |
---|

Complexity theory | $~$17$~$ |
---|

Complexity theory | $~$13 \cdot 17 = 221$~$ |
---|

Complexity theory: Complexity zoo | $~$P$~$ |
---|

Complexity theory: Complexity zoo | $~$x$~$ |
---|

Complexity theory: Complexity zoo | $~$1000 x^{42}+10^{100}$~$ |
---|

Complexity theory: Complexity zoo | $~$P$~$ |
---|

Complexity theory: Complexity zoo | $~$\mathcal{O}(n)$~$ |
---|

Complexity theory: Complexity zoo | $~$\mathcal{O}(n*log(n))$~$ |
---|

Complexity theory: Complexity zoo | $~$P$~$ |
---|

Complexity theory: Complexity zoo | $~$NP$~$ |
---|

Complexity theory: Complexity zoo | $~$NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P$~$ |
---|

Complexity theory: Complexity zoo | $~$P$~$ |
---|

Complexity theory: Complexity zoo | $~$NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P\subset NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P=NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P!=NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P!=NP$~$ |
---|

Complexity theory: Complexity zoo | $~$P=NP$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Compressing multiple messages | $~$\lceil \log_2(n) \rceil$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Compressing multiple messages | $~$3^{10} < 2^{16}.$~$ |
---|

Compressing multiple messages | $~$3^{10}$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Compressing multiple messages | $~$k$~$ |
---|

Compressing multiple messages | $~$n^k$~$ |
---|

Compressing multiple messages | $~$n^k$~$ |
---|

Compressing multiple messages | $~$k$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Compressing multiple messages | $~$k$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Compressing multiple messages | $~$n$~$ |
---|

Concrete groups (Draft) | $~$1$~$ |
---|

Concrete groups (Draft) | $~$2$~$ |
---|

Concrete groups (Draft) | $~$3$~$ |
---|

Concrete groups (Draft) | $~$4$~$ |
---|

Concrete groups (Draft) | $~$90^\circ$~$ |
---|

Concrete groups (Draft) | $~$1 \mapsto 2$~$ |
---|

Concrete groups (Draft) | $~$2 \mapsto 3$~$ |
---|

Concrete groups (Draft) | $~$3 \mapsto 4$~$ |
---|

Concrete groups (Draft) | $~$4 \mapsto 1$~$ |
---|

Concrete groups (Draft) | $~$r := (1234)$~$ |
---|

Concrete groups (Draft) | $~$r^2 = (13)(24)$~$ |
---|

Concrete groups (Draft) | $~$180^\circ$~$ |
---|

Concrete groups (Draft) | $~$r^3 = (4321)$~$ |
---|

Concrete groups (Draft) | $~$270^\circ$~$ |
---|

Concrete groups (Draft) | $~$f:= (1 4)(2 3)$~$ |
---|

Concrete groups (Draft) | $~$180^\circ$~$ |
---|

Concrete groups (Draft) | $~$(13)(24)\circ(14)(23) = (1 2)(3 4)$~$ |
---|

Concrete groups (Draft) | $~$f$~$ |
---|

Concrete groups (Draft) | $~$r$~$ |
---|

Concrete groups (Draft) | $~$rf = (1234)(14)(23)$~$ |
---|

Concrete groups (Draft) | $~$(13) = r^3f$~$ |
---|

Concrete groups (Draft) | $~$90^\circ$~$ |
---|

Concrete groups (Draft) | $~$270^\circ$~$ |
---|

Concrete groups (Draft) | $~$(24)(24) = ()$~$ |
---|

Concrete groups (Draft) | $~$(4321)(1234) = ()$~$ |
---|

Concrete groups (Draft) | $~$r$~$ |
---|

Concrete groups (Draft) | $~$r^2$~$ |
---|

Concrete groups (Draft) | $~$r^3$~$ |
---|

Concrete groups (Draft) | $~$f$~$ |
---|

Concrete groups (Draft) | $~$rf$~$ |
---|

Concrete groups (Draft) | $~$r^2f$~$ |
---|

Concrete groups (Draft) | $~$r^3f$~$ |
---|

Concrete groups (Draft) | $~$e := ()$~$ |
---|

Concrete groups (Draft) | $~$(12)$~$ |
---|

Concrete groups (Draft) | $~$G$~$ |
---|

Concrete groups (Draft) | $~$\circ : G \times G \to G$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|

Conditional probability | $~$\mathbb{P}(yellow\mid banana)$~$ |
---|

Conditional probability | $~$\mathbb{P}(banana\mid yellow)$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|

Conditional probability | $~$\mathbb{P}(yellow\mid banana)$~$ |
---|

Conditional probability | $~$\mathbb{P}(banana\mid yellow)$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|

Conditional probability | $~$\mathbb{P}(blue \wedge round)$~$ |
---|

Conditional probability | $~$\mathbb{P}(blue\mid round) := \frac{\mathbb{P}(blue \wedge round)}{\mathbb{P}(round)} = \frac{\text{5% blue and round marbles}}{\text{20% round marbles}} = \frac{5}{20} = 0.25.$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y) := \frac{\mathbb{P}(X \wedge Y)}{\mathbb{P}(Y)}.$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y) := \frac{\mathbb{P}(X \wedge Y)}{\mathbb{P}(Y)}$~$ |
---|

Conditional probability | $~$Y$~$ |
---|

Conditional probability | $~$X$~$ |
---|

Conditional probability | $~$Y$~$ |
---|

Conditional probability | $~$X \wedge Y$~$ |
---|

Conditional probability | $~$X \wedge Y$~$ |
---|

Conditional probability | $~$\mathbb P(observation\mid hypothesis)$~$ |
---|

Conditional probability | $~$\mathbb P(hypothesis\mid observation)$~$ |
---|

Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|

Conditional probability | $~$X$~$ |
---|

Conditional probability | $~$Y$~$ |
---|

Conditional probability | $~$\mathbb P(left\mid right)$~$ |
---|

Conditional probability | $~$left$~$ |
---|

Conditional probability | $~$right$~$ |
---|

Conditional probability | $~$\mathbb P(yellow\mid banana)$~$ |
---|

Conditional probability | $~$\mathbb P(banana\mid yellow)$~$ |
---|

Conditional probability | $~$yellow$~$ |
---|

Conditional probability | $~$banana$~$ |
---|

Conditional probability | $~$\mathbb P(left \mid right),$~$ |
---|

Conditional probability | $~$right$~$ |
---|

Conditional probability | $~$right$~$ |
---|

Conditional probability | $~$left$~$ |
---|

Conditional probability | $~$X \wedge Y$~$ |
---|

Conditional probability | $~$X$~$ |
---|

Conditional probability | $~$Y$~$ |
---|

Conditional probability | $~$X$~$ |
---|

Conditional probability | $~$Y$~$ |
---|

Conditional probability | $$~$\mathbb P(left \mid right) = \dfrac{\mathbb P(left \wedge right)}{\mathbb P(right)}.$~$$ |
---|

Conditional probability | $~$right$~$ |
---|

Conditional probability | $~$right$~$ |
---|

Conditional probability | $~$left$~$ |
---|

Conditional probability | $$~$\begin{array}{l\mid r\mid r}
& Red & Blue \\
\hline
Square & 1 & 2 \\
\hline
Round & 3 & 4
\end{array}$~$$ |
---|

Conditional probability | $$~$\mathbb P(red\mid round) = \dfrac{\mathbb P(red \wedge round)}{\mathbb P(round)} = \dfrac{3}{3 + 4} = \dfrac{3}{7}$~$$ |
---|

Conditional probability | $$~$\mathbb P(square\mid blue) = \dfrac{\mathbb P(square \wedge blue)}{\mathbb P(blue)} = \dfrac{2}{2 + 4} = \dfrac{1}{3}$~$$ |
---|

Conditional probability | $~$\mathbb P(red hair\mid Scarlet) = 99\%,$~$ |
---|

Conditional probability | $~$\mathbb P(redhair\mid Scarlet),$~$ |
---|

Conditional probability | $~$\mathbb P(Scarlet\mid redhair),$~$ |
---|

Conditional probability | $~$\mathbb P(redhair\mid Scarlet)$~$ |
---|

Conditional probability | $~$1$~$ |
---|

Conditional probability | $~$\mathbb P(redhair\mid Scarlet)$~$ |
---|

Conditional probability | $~$\mathbb P(Scarlet\mid redhair)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|

Conditional probability: Refresher | $~$\frac{\mathbb P(\text{left} \land \text{right})}{\mathbb P(\text{right})}.$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(yellow \mid banana)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(banana \mid yellow)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|

Conditional probability: Refresher | $~$\frac{\mathbb P(\text{left} \land \text{right})}{\mathbb P(\text{right})}.$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(yellow \mid banana)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(banana \mid yellow)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(v)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(V = v)$~$ |
---|

Conditional probability: Refresher | $~$V$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(yellow)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P({ColorOfNextObjectInBag}=yellow)$~$ |
---|

Conditional probability: Refresher | $~$ColorOfNextObjectInBag$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P,$~$ |
---|

Conditional probability: Refresher | $~$yellow$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(x \land y)$~$ |
---|

Conditional probability: Refresher | $~$x$~$ |
---|

Conditional probability: Refresher | $~$y$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(x\mid y)$~$ |
---|

Conditional probability: Refresher | $$~$\frac{\mathbb P(x \wedge y)}{\mathbb P(y)}.$~$$ |
---|

Conditional probability: Refresher | $~$\mathbb P({sick}\mid {positive})$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P({sick}\mid {positive})$~$ |
---|

Conditional probability: Refresher | $~$=$~$ |
---|

Conditional probability: Refresher | $~$\frac{\mathbb P({sick} \land {positive})}{\mathbb P({positive})}.$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(sick \mid positive)$~$ |
---|

Conditional probability: Refresher | $~$sick$~$ |
---|

Conditional probability: Refresher | $~$positive$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(x\mid y)$~$ |
---|

Conditional probability: Refresher | $~$y$~$ |
---|

Conditional probability: Refresher | $~$y$~$ |
---|

Conditional probability: Refresher | $~$x$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(positive \mid sick)$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(sick \mid positive).$~$ |
---|

Conditional probability: Refresher | $~$\frac{18}{20} = 0.9$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(positive \mid sick) = 90\%,$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(sick \mid positive) \approx 43\%.$~$ |
---|

Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|

Conjugacy class | $~$g$~$ |
---|

Conjugacy class | $~$G$~$ |
---|

Conjugacy class | $~$g$~$ |
---|

Conjugacy class | $~$G$~$ |
---|

Conjugacy class | $~$\{ x g x^{-1} : x \in G \}$~$ |
---|

Conjugacy class | $~$g$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$S_n$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$S_n$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$n_1, \dots, n_k$~$ |
---|

Conjugacy class is cycle type in symmetric group | $$~$\sigma = (a_{11} a_{12} \dots a_{1 n_1})(a_{21} \dots a_{2 n_2}) \dots (a_{k 1} a_{k 2} \dots a_{k n_k})$~$$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\tau \in S_n$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\tau \sigma \tau^{-1}(\tau(a_{ij})) = \tau \sigma(a_{ij}) = a_{i (j+1)}$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$a_{i (n_i+1)}$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$a_{i 1}$~$ |
---|

Conjugacy class is cycle type in symmetric group | $$~$\tau \sigma \tau^{-1} = (\tau(a_{11}) \tau(a_{12}) \dots \tau(a_{1 n_1}))(\tau(a_{21}) \dots \tau(a_{2 n_2})) \dots (\tau(a_{k 1}) \tau(a_{k 2}) \dots \tau(a_{k n_k}))$~$$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|

Conjugacy class is cycle type in symmetric group | $$~$\pi = (b_{11} b_{12} \dots b_{1 n_1})(b_{21} \dots b_{2 n_2}) \dots (b_{k 1} b_{k 2} \dots b_{k n_k})$~$$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\pi$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\tau(a_{ij}) = b_{ij}$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\tau$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\tau \sigma \tau^{-1} = \pi$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$\pi$~$ |
---|

Conjugacy class is cycle type in symmetric group | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$5!/2 = 60$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(5)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(3, 1, 1)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(2, 2, 1)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(1,1,1,1,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(5)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(12345)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(12345)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements | $~$(12)(12345)(12)^{-1} = (21345)$~$ |
---|

Conjugacy classes of the alternating group on five elements | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 12 & 5 & 5 \\ \hline
(21345) & 12 & 5 & 5 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$60$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5! = 120$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 12 & 5 & 5 \\ \hline
(21345) & 12 & 5 & 5 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau e \tau^{-1} = \tau \tau^{-1} = e$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_n$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_n$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(5)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(3,1,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(2,2,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(1,1,1,1,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(2,2,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(ce)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(de)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ac)(bd)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(cba)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$e$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ac)(be)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bc)(de)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$e$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(3,1,1)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(acb)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bc)(de)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abd)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(cde)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ade)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bd)(ce)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\{ \rho (12345) \rho^{-1}: \rho \ \text{even} \}$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\{ \rho (12345) \rho^{-1}: \rho \ \text{odd} \}$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345) = (12)(12345)(12)^{-1}$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau (12345) \tau^{-1} = (\tau(1), \tau(2), \tau(3), \tau(4), \tau(5))$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$1$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$2$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$2$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$1$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$3$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$3$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$4$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$4$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|

Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5! = 120$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$S_5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 24 & 5 & 5 \\ \hline
(1234) & 30 & 4,1 & 4 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(123)(45) & 20 & 3,2 & 6 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
(12) & 10 & 2,1,1,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$6$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12345)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12345)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(23451)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(34512)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4!$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$24$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(1234)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4 \times 3 \times 2 = 24$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3 \times 2 \times 1 = 6$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$30$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3,1,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3,2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3,1,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(123)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$4,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{3} = 10$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\{1,2,3\}$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(123)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(231)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(312)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(132)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(321)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(213)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2 \times 10 = 20$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3,2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(123)(45)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{3} = 10$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2 \times 10 = 20$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2,2,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2,1,1,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2,2,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)(34)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2}$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{3}{2}$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)(34)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(34)(12)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2} \times \binom{3}{2} / 2 = 15$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2,1,1,1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2}$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2} = 10$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|

Conjunctions and disjunctions | $~$P \land Q$~$ |
---|

Conjunctions and disjunctions | $~$P \lor Q$~$ |
---|

Conjunctions and disjunctions | $~$R$~$ |
---|

Conjunctions and disjunctions | $~$P$~$ |
---|

Conjunctions and disjunctions | $~$Q$~$ |
---|

Conjunctions and disjunctions | $~$R \equiv P \land Q $~$ |
---|

Conjunctions and disjunctions | $~$S$~$ |
---|

Conjunctions and disjunctions | $~$P$~$ |
---|

Conjunctions and disjunctions | $~$Q$~$ |
---|

Conjunctions and disjunctions | $~$S$~$ |
---|

Conjunctions and disjunctions | $~$P$~$ |
---|

Conjunctions and disjunctions | $~$Q$~$ |
---|

Conjunctions and disjunctions | $~$S \equiv P \lor Q$~$ |
---|

Consequentialist cognition | $~$X$~$ |
---|

Consequentialist cognition | $~$X$~$ |
---|

Consequentialist cognition | $~$Y$~$ |
---|

Consequentialist cognition | $~$Y$~$ |
---|

Consequentialist cognition | $~$Y'$~$ |
---|

Consequentialist cognition | $~$X$~$ |
---|

Consequentialist cognition | $~$X',$~$ |
---|

Consequentialist cognition | $~$X$~$ |
---|

Consequentialist cognition | $~$Y$~$ |
---|

Consistency | $~$X$~$ |
---|

Consistency | $~$T\vdash X$~$ |
---|

Consistency | $~$T\vdash \neg X$~$ |
---|

Consistency | $~$\square_{PA}$~$ |
---|

Consistency | $~$\neg\square_{PA}(\ulcorner 0=1\urcorner)$~$ |
---|

Consistency | $~$PA$~$ |
---|

Consistency | $~$PA$~$ |
---|

Context disaster | $~$V$~$ |
---|

Context disaster | $~$V$~$ |
---|

Context disaster | $~$0$~$ |
---|

Context disaster | $~$0,$~$ |
---|

Context disaster | $~$V$~$ |
---|

Context disaster | $~$0$~$ |
---|

Context disaster | $~$U$~$ |
---|

Context disaster | $~$\mathbb P_t(X)$~$ |
---|

Context disaster | $~$X$~$ |
---|

Context disaster | $~$t,$~$ |
---|

Context disaster | $~$\mathbb Q_t(X)$~$ |
---|

Context disaster | $~$X$~$ |
---|

Context disaster | $~$\pi \in \Pi$~$ |
---|

Context disaster | $~$\pi$~$ |
---|

Context disaster | $~$\Pi$~$ |
---|

Context disaster | $~$\mathbb E_{\mathbb P, t} [W \mid \pi]$~$ |
---|

Context disaster | $~$\mathbb P_t$~$ |
---|

Context disaster | $~$W$~$ |
---|

Context disaster | $~$\pi$~$ |
---|

Context disaster | $$~$\underset{\pi \in \Pi}{\operatorname {optimum}} F(\pi)$~$$ |
---|

Context disaster | $~$\pi$~$ |
---|

Context disaster | $~$\Pi$~$ |
---|

Context disaster | $~$F$~$ |
---|

Context disaster | $~$\Pi_1$~$ |
---|

Context disaster | $~$t,$~$ |
---|

Context disaster | $~$\Pi_2$~$ |
---|

Context disaster | $~$u$~$ |
---|

Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, u} [V \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U \mid \pi] \big ] < 0$~$$ |
---|

Context disaster | $~$t$~$ |
---|

Context disaster | $~$\Pi_1$~$ |
---|

Context disaster | $~$V$~$ |
---|

Context disaster | $~$u$~$ |
---|

Context disaster | $~$\Pi_2,$~$ |
---|

Context disaster | $~$V.$~$ |
---|

Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] < 0 \\
\mathbb E_{\mathbb P, u} [V \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U \mid \pi] \big ] < 0$~$$ |
---|

Context disaster | $~$V.$~$ |
---|

Context disaster | $~$W_{t}$~$ |
---|

Context disaster | $~$W$~$ |
---|

Context disaster | $~$t,$~$ |
---|

Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V_\infty \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V_{u} \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V_\infty \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] < 0 \\
\mathbb E_{\mathbb P, u} [V_\infty \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U_\infty \mid \pi] \big ] < 0$~$$ |
---|

Context disaster | $~$t$~$ |
---|

Context disaster | $~$u$~$ |
---|

Context disaster | $~$V,$~$ |
---|

Context disaster | $~$t$~$ |
---|

Context disaster | $~$\mathbb Q_t$~$ |
---|

Context disaster | $~$V$~$ |
---|

Context disaster | $~$U,$~$ |
---|

Context disaster | $~$U$~$ |
---|

Context disaster | $~$U$~$ |
---|

Context disaster | $~$V,$~$ |
---|

Context disaster | $~$U$~$ |
---|

Context disaster | $~$V.$~$ |
---|

Convergent instrumental strategies | $~$X$~$ |
---|

Convergent instrumental strategies | $~$X$~$ |
---|

Convergent instrumental strategies | $~$X,$~$ |
---|

Convergent instrumental strategies | $~$X'$~$ |
---|

Convergent instrumental strategies | $~$X$~$ |
---|

Convergent instrumental strategies | $~$X'$~$ |
---|

Convergent instrumental strategies | $~$X^*$~$ |
---|

Convergent instrumental strategies | $~$\pi_1$~$ |
---|

Convergent instrumental strategies | $~$\pi_2$~$ |
---|

Convergent strategies of self-modification | $~$X$~$ |
---|

Convergent strategies of self-modification | $~$Y.$~$ |
---|

Convergent strategies of self-modification | $~$Y$~$ |
---|

Convergent strategies of self-modification | $~$X$~$ |
---|

Convergent strategies of self-modification | $~$Y.$~$ |
---|

Convergent strategies of self-modification | $~$Y$~$ |
---|

Convergent strategies of self-modification | $~$X$~$ |
---|

Convergent strategies of self-modification | $~$X$~$ |
---|

Convergent strategies of self-modification | $~$Y.$~$ |
---|

Convex set | $~$x$~$ |
---|

Convex set | $~$y$~$ |
---|

Convex set | $~$x$~$ |
---|

Convex set | $~$y$~$ |
---|

Convex set | $~$S$~$ |
---|

Convex set | $$~$\forall x, y \in S, \theta \in [0, 1]: \theta x + (1 - \theta) y \in S$~$$ |
---|

Cosmic endowment | $~$\approx 4 \times 10^{20}$~$ |
---|

Cosmic endowment | $~$\approx 10^{42}$~$ |
---|

Cosmic endowment | $~$\approx 10^{25}$~$ |
---|

Cosmic endowment | $~$\approx 10^{54}$~$ |
---|

Countability | $~$\mathbb{Z}^+ = \{1, 2, 3, 4, \ldots\}$~$ |
---|

Countability | $~$S$~$ |
---|

Countability | $~$S$~$ |
---|

Countability | $~$\mathbb Q$~$ |
---|

Countability | $~$\frac{p}{q}$~$ |
---|

Countability | $~$p$~$ |
---|

Countability | $~$q$~$ |
---|

Countability | $~$q > 0$~$ |
---|

Countability | $~$\mathbb Z^+ \times \mathbb Z^+$~$ |
---|

Countability | $~$\mathbb Z$~$ |
---|

Countability | $~$\frac{a}{b}$~$ |
---|

Countability | $~$|a| + |b|$~$ |
---|

Countability | $~$a$~$ |
---|

Countability | $~$b$~$ |
---|

Countability | $~$0 / 1$~$ |
---|

Countability | $~$-1 / 1$~$ |
---|

Countability | $~$1 / 1$~$ |
---|

Countability | $~$-2 / 1$~$ |
---|

Countability | $~$-1 / 2$~$ |
---|

Countability | $~$1 / 2$~$ |
---|

Countability | $~$2 / 1$~$ |
---|

Countability | $~$\ldots$~$ |
---|

Countability | $~$(2d+1)^2$~$ |
---|

Countability | $~$d$~$ |
---|

Countability | $~$d$~$ |
---|

Countability | $~$(2d+1)^2$~$ |
---|

Countability | $~$\square$~$ |
---|

Countability | $~$(\mathbb Z^+)^n$~$ |
---|

Countability | $~$n$~$ |
---|

Countability | $~$f$~$ |
---|

Countability | $~$A$~$ |
---|

Countability | $~$B$~$ |
---|

Countability | $~$B$~$ |
---|

Countability | $~$E$~$ |
---|

Countability | $~$A$~$ |
---|

Countability | $~$E\circ f$~$ |
---|

Countability | $~$B$~$ |
---|

Countability | $~$B$~$ |
---|

Countability | $~$\Sigma^*$~$ |
---|

Countability | $~$\mathbb N^n$~$ |
---|

Countability | $~$n$~$ |
---|

Countability | $~$n\in \mathbb N$~$ |
---|

Countability | $~$E_n: \mathbb N \to \mathbb N^n$~$ |
---|

Countability | $~$\mathbb N ^n$~$ |
---|

Countability | $~$(J_1,J_2)(n)$~$ |
---|

Countability | $~$\mathbb N^2$~$ |
---|

Countability | $~$E: \mathbb N \to \Sigma^* , n\hookrightarrow E_{J_1(n)}(J_2(n))$~$ |
---|

Countability | $~$\Sigma^*$~$ |
---|

Countability | $~$E$~$ |
---|

Countability | $~$\Sigma^*$~$ |
---|

Countability | $~$\square$~$ |
---|

Countability | $~$P_\omega(A)$~$ |
---|

Countability | $~$A$~$ |
---|

Countability | $~$E$~$ |
---|

Countability | $~$A$~$ |
---|

Countability | $~$E': \mathbb N^* \to P_\omega(A)$~$ |
---|

Countability | $~$n_0 n_1 n_2 … n_r$~$ |
---|

Countability | $~$\{a\in A:\exists m\le k E(n_m)=a\}\subseteq A$~$ |
---|

Countability | $~$E'$~$ |
---|

Countability | $~$P_\omega(A)$~$ |
---|

Creating a /learn/ link | $~$bayes_rule_details,$~$ |
---|

Currying | $~$F:(X,Y,Z,N)→R$~$ |
---|

Currying | $~$curry(F)$~$ |
---|

Currying | $~$X→(Y→(Z→(N→R)))$~$ |
---|

Currying | $~$curry(F)(4)(3)(2)(1)$~$ |
---|

Currying | $~$F(4,3,2,1)$~$ |
---|

Cycle notation in symmetric groups | $~$k$~$ |
---|

Cycle notation in symmetric groups | $~$k$~$ |
---|

Cycle notation in symmetric groups | $~$S_n$~$ |
---|

Cycle notation in symmetric groups | $~$k$~$ |
---|

Cycle notation in symmetric groups | $~$a_1, \dots, a_k$~$ |
---|

Cycle notation in symmetric groups | $~$\{1,2,\dots,n\}$~$ |
---|

Cycle notation in symmetric groups | $~$k$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma(a_i) = a_{i+1}$~$ |
---|

Cycle notation in symmetric groups | $~$1 \leq i < k$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma(a_k) = a_1$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma(x) = x$~$ |
---|

Cycle notation in symmetric groups | $~$x \not \in \{a_1, \dots, a_k \}$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma = (a_1 a_2 \dots a_k)$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma = (a_1, a_2, \dots, a_k)$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k) = (a_2 a_3 \dots a_k a_1)$~$ |
---|

Cycle notation in symmetric groups | $~$a_i$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Cycle notation in symmetric groups | $~$(a_k a_{k-1} \dots a_1)$~$ |
---|

Cycle notation in symmetric groups | $$~$\begin{pmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix}$~$$ |
---|

Cycle notation in symmetric groups | $~$(123)$~$ |
---|

Cycle notation in symmetric groups | $~$(231)$~$ |
---|

Cycle notation in symmetric groups | $~$(312)$~$ |
---|

Cycle notation in symmetric groups | $~$(123)$~$ |
---|

Cycle notation in symmetric groups | $~$S_n$~$ |
---|

Cycle notation in symmetric groups | $~$n \geq 3$~$ |
---|

Cycle notation in symmetric groups | $~$(145)$~$ |
---|

Cycle notation in symmetric groups | $~$S_n$~$ |
---|

Cycle notation in symmetric groups | $~$n \geq 5$~$ |
---|

Cycle notation in symmetric groups | $~$S_n$~$ |
---|

Cycle notation in symmetric groups | $~$S_4$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $$~$\begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ \end{pmatrix}$~$$ |
---|

Cycle notation in symmetric groups | $~$(12)$~$ |
---|

Cycle notation in symmetric groups | $~$(34)$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$c_1 = (a_1 a_2 \dots a_k)$~$ |
---|

Cycle notation in symmetric groups | $~$c_2$~$ |
---|

Cycle notation in symmetric groups | $~$c_3$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma = c_3 c_2 c_1$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Cycle notation in symmetric groups | $~$a_1 \mapsto a_2 \mapsto a_3 \dots \mapsto a_k \mapsto a_1$~$ |
---|

Cycle notation in symmetric groups | $~$k$~$ |
---|

Cycle notation in symmetric groups | $~$i$~$ |
---|

Cycle notation in symmetric groups | $~$a_1 \mapsto a_{i+1}$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)(a_4 a_5)$~$ |
---|

Cycle notation in symmetric groups | $~$a_i$~$ |
---|

Cycle notation in symmetric groups | $~$3 \times 2 = 6$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)$~$ |
---|

Cycle notation in symmetric groups | $~$(a_4 a_5)$~$ |
---|

Cycle notation in symmetric groups | $~$[(a_1 a_2 a_3)(a_4 a_5)]^n = (a_1 a_2 a_3)^n (a_4 a_5)^n$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n (a_4 a_5)^n$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n = (a_4 a_5)^n = e$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n$~$ |
---|

Cycle notation in symmetric groups | $~$a_1$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n$~$ |
---|

Cycle notation in symmetric groups | $~$n$~$ |
---|

Cycle notation in symmetric groups | $~$3$~$ |
---|

Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)$~$ |
---|

Cycle notation in symmetric groups | $~$3$~$ |
---|

Cycle notation in symmetric groups | $~$(a_4 a_5)^n$~$ |
---|

Cycle notation in symmetric groups | $~$n$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$S_5$~$ |
---|

Cycle notation in symmetric groups | $~$(123)$~$ |
---|

Cycle notation in symmetric groups | $~$(345)$~$ |
---|

Cycle notation in symmetric groups | $~$(345)(123) = (12453)$~$ |
---|

Cycle notation in symmetric groups | $~$1$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$1$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$3$~$ |
---|

Cycle notation in symmetric groups | $~$3$~$ |
---|

Cycle notation in symmetric groups | $~$4$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$2$~$ |
---|

Cycle notation in symmetric groups | $~$4$~$ |
---|

Cycle notation in symmetric groups | $~$4$~$ |
---|

Cycle notation in symmetric groups | $~$4$~$ |
---|

Cycle notation in symmetric groups | $~$5$~$ |
---|

Cycle notation in symmetric groups | $~$\sigma$~$ |
---|

Cycle notation in symmetric groups | $~$4$~$ |
---|

Cycle notation in symmetric groups | $~$5$~$ |
---|

Cycle type of a permutation | $~$\sigma$~$ |
---|

Cycle type of a permutation | $~$S_n$~$ |
---|

Cycle type of a permutation | $~$\sigma$~$ |
---|

Cycle type of a permutation | $~$\sigma$~$ |
---|

Cycle type of a permutation | $~$\sigma$~$ |
---|

Cycle type of a permutation | $~$1$~$ |
---|

Cycle type of a permutation | $~$(123)(45)$~$ |
---|

Cycle type of a permutation | $~$S_7$~$ |
---|

Cycle type of a permutation | $~$3,2$~$ |
---|

Cycle type of a permutation | $~$(6)$~$ |
---|

Cycle type of a permutation | $~$(7)$~$ |
---|

Cycle type of a permutation | $~$3,2,1,1$~$ |
---|

Cycle type of a permutation | $~$k$~$ |
---|

Cycle type of a permutation | $~$k$~$ |
---|

Cycle type of a permutation | $~$k$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$6$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$6$~$ |
---|

Cyclic Group Intro (Math 0) | $~$11$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$9$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$9$~$ |
---|

Cyclic Group Intro (Math 0) | $~$4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7+9 = 16$~$ |
---|

Cyclic Group Intro (Math 0) | $~$16- 12 = 4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$4 + 12 = 16$~$ |
---|

Cyclic Group Intro (Math 0) | $~$16 - 12 = 4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12 - 5 = 7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$2$~$ |
---|

Cyclic Group Intro (Math 0) | $~$6$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$9$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7+9 = 16$~$ |
---|

Cyclic Group Intro (Math 0) | $~$16-12 = 4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7 +5 = 12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12 - 12 = 0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$\bullet$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7 \bullet 9 = 4$~$ |
---|

Cyclic Group Intro (Math 0) | $~$12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5 \bullet 7 = 12$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7 \bullet 9 = 1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7 + 9 = 16$~$ |
---|

Cyclic Group Intro (Math 0) | $~$16 - 15 = 1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15 - 5 = 10$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$10$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5 + 10 = 15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5 \bullet 10 = 0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-5 = 10$~$ |
---|

Cyclic Group Intro (Math 0) | $~$5$~$ |
---|

Cyclic Group Intro (Math 0) | $~$7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-5 = 7$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1 \bullet 1 \bullet 1 \bullet \cdots \bullet 1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-1 = 11$~$ |
---|

Cyclic Group Intro (Math 0) | $~$15$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-1 = 14$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$h$~$ |
---|

Cyclic Group Intro (Math 0) | $~$t$~$ |
---|

Cyclic Group Intro (Math 0) | $~$\bullet$~$ |
---|

Cyclic Group Intro (Math 0) | $~$h \bullet h = t$~$ |
---|

Cyclic Group Intro (Math 0) | $~$h \bullet t = h$~$ |
---|

Cyclic Group Intro (Math 0) | $~$t \bullet h = h$~$ |
---|

Cyclic Group Intro (Math 0) | $~$t \bullet t = t$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1 \bullet 1 = 0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1 \bullet 0 = 1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0 \bullet 1 = 1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$0 \bullet 0 = 0$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic Group Intro (Math 0) | $~$1$~$ |
---|

Cyclic group | $~$G$~$ |
---|

Cyclic group | $~$g$~$ |
---|

Cyclic group | $~$g$~$ |
---|

Cyclic group | $~$(G, +)$~$ |
---|

Cyclic group | $~$G$~$ |
---|

Cyclic group | $~$g \in G$~$ |
---|

Cyclic group | $~$h \in G$~$ |
---|

Cyclic group | $~$n \in \mathbb{Z}$~$ |
---|

Cyclic group | $~$h = g^n$~$ |
---|

Cyclic group | $~$g^n$~$ |
---|

Cyclic group | $~$g + g + \dots + g$~$ |
---|

Cyclic group | $~$n$~$ |
---|

Cyclic group | $~$G = \langle g \rangle$~$ |
---|

Cyclic group | $~$g$~$ |
---|

Cyclic group | $~$G$~$ |
---|

Cyclic group | $~$(\mathbb{Z}, +) = \langle 1 \rangle = \langle -1 \rangle$~$ |
---|

Cyclic group | $~$\{ e, g \}$~$ |
---|

Cyclic group | $~$e$~$ |
---|

Cyclic group | $~$g^2 = e$~$ |
---|

Cyclic group | $~$g$~$ |
---|

Cyclic group | $~$g^2 = g^0 = e$~$ |
---|

Cyclic group | $~$n$~$ |
---|

Cyclic group | $~$n$~$ |
---|

Cyclic group | $~$1$~$ |
---|

Cyclic group | $~$n-1$~$ |
---|

Cyclic group | $~$S_n$~$ |
---|

Cyclic group | $~$n > 2$~$ |
---|

Cyclic group | $~$a, b \in G$~$ |
---|

Cyclic group | $~$g$~$ |
---|

Cyclic group | $~$G$~$ |
---|

Cyclic group | $~$a = g^i, b = g^j$~$ |
---|

Cyclic group | $~$ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$~$ |
---|

Cyclic group | $~$\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \dots \}$~$ |
---|

Data capacity | $~$\log(2)$~$ |
---|

Data capacity | $~$\log_2(2)=1$~$ |
---|

Data capacity | $~$\log_2(36) \approx 5.17$~$ |
---|

Data capacity | $~$\log_2(8) = 3$~$ |
---|

Data capacity | $~$n$~$ |
---|

Data capacity | $~$b$~$ |
---|

Data capacity | $~$b^n$~$ |
---|

Data capacity | $~$5 \cdot 8 = 40$~$ |
---|

Death in Damascus | $~$\operatorname {do}()$~$ |
---|

Death in Damascus | $~$D$~$ |
---|

Death in Damascus | $~$A$~$ |
---|

Death in Damascus | $~$Y$~$ |
---|

Death in Damascus | $~$N$~$ |
---|

Death in Damascus | $~$DY, AY, DN, AN$~$ |
---|

Death in Damascus | $$~$
\begin{array}{r|c|c}
& \text {Damascus fatal} & \text {Aleppo fatal} \\ \hline
\ {DN} & \text {Die} & \text{Live} \\ \hline
\ {AN} & \text {Live} & \text {Die} \\ \hline
\ {DY} & \text {Die, \$-1} & \text{Live, \$+10} \\ \hline
\ {AY} & \text {Live, \$+10} & \text {Die, \$-1}
\end{array}
$~$$ |
---|

Death in Damascus | $~$AY$~$ |
---|

Death in Damascus | $~$AN.$~$ |
---|

Decimal notation | $~$e$~$ |
---|

Decimal notation | $~$(2 \cdot 100) + (4 \cdot 10) + (6 \cdot 1),$~$ |
---|

Decision problem | $~$w$~$ |
---|

Decision problem | $~$p$~$ |
---|

Decision problem | $~$D$~$ |
---|

Decision problem | $~$A$~$ |
---|

Decision problem | $~$A$~$ |
---|

Decision problem | $~$\{0,1\}^*$~$ |
---|

Decision problem | $~$w$~$ |
---|

Decision problem | $~$p$~$ |
---|

Decision problem | $~$w$~$ |
---|

Decision problem | $~$A$~$ |
---|

Decision problem | $~$w$~$ |
---|

Decision problem | $~$D$~$ |
---|

Decision problem | $~$D$~$ |
---|

Decision problem | $~$A$~$ |
---|

Decision problem | $~$D$~$ |
---|

Decision problem | $~$D$~$ |
---|

Decision problem | $$~$
CONNECTED = \{s\in\{0,1\}^*:\text{$s$ represents a connected graph}\}
$~$$ |
---|

Decision problem | $~$TAUTOLOGY$~$ |
---|

Decision problem | $~$TAUTOLOGY$~$ |
---|

Decision problem | $$~$
PRIMES = \{ x\in \mathbb{N}:\text{$x$ is prime}\}
$~$$ |
---|

Decision problem | $~$PRIMES$~$ |
---|

Decision problem | $$~$
PRIMES = \{s\in\{0,1\}^*:\text{$s$ represent a prime number in base $2$}\}
$~$$ |
---|

Decit | $~$\log_2(10)\approx 3.32$~$ |
---|

Dependent messages can be encoded cheaply | $~$m_1, m_2, m_3$~$ |
---|

Dependent messages can be encoded cheaply | $~$E$~$ |
---|

Dependent messages can be encoded cheaply | $~$E(m_1)E(m_2)E(m_3)$~$ |
---|

Dependent messages can be encoded cheaply | $~$(m_1, m_2, m_3)$~$ |
---|

Derivative | $~$y$~$ |
---|

Derivative | $~$x$~$ |
---|

Derivative | $~$y$~$ |
---|

Derivative | $~$x$~$ |
---|

Derivative | $~$f(x)$~$ |
---|

Derivative | $~$x$~$ |
---|

Derivative | $~$f(x)$~$ |
---|

Derivative | $~$(x, f(x))$~$ |
---|

Derivative | $~$t = 0$~$ |
---|

Derivative | $~$4.7 t^2$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $$~$\frac{\mathrm{d}}{\mathrm{d} t} mileage = speed$~$$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$4.7 t^2$~$ |
---|

Derivative | $$~$\frac{\mathrm{d}}{\mathrm{d} t} 4.7 t^2 = speed$~$$ |
---|

Derivative | $$~$distance\ traveled = 2t$~$$ |
---|

Derivative | $~$distance\ traveled = 2t$~$ |
---|

Derivative | $~$distance\ traveled = t^2$~$ |
---|

Derivative | $~$t=1$~$ |
---|

Derivative | $~$d = t^2$~$ |
---|

Derivative | $~$d$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$\frac{\Delta d}{\Delta t}$~$ |
---|

Derivative | $~$(t,t^2)$~$ |
---|

Derivative | $~$h$~$ |
---|

Derivative | $~$((t+h),(t+h)^2)$~$ |
---|

Derivative | $$~$∆d=(t+h)^2-t^2$~$$ |
---|

Derivative | $$~$∆t=(t+h) - t$~$$ |
---|

Derivative | $$~$∆d=2ht + h^2$~$$ |
---|

Derivative | $$~$∆t=h$~$$ |
---|

Derivative | $$~$\frac{\Delta d}{\Delta t}=\frac{2ht + h^2}{h}=2t+h$~$$ |
---|

Derivative | $~$h$~$ |
---|

Derivative | $~$2t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$1$~$ |
---|

Derivative | $~$2$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$5$~$ |
---|

Derivative | $~$10$~$ |
---|

Derivative | $~$t^2$~$ |
---|

Derivative | $~$2t$~$ |
---|

Derivative | $~$4.7t^2$~$ |
---|

Derivative | $~$9.4t$~$ |
---|

Derivative | $~$t=0$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$9.4t$~$ |
---|

Derivative | $~$t^2$~$ |
---|

Derivative | $~$2t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$t$~$ |
---|

Derivative | $~$c$~$ |
---|

Derivative | $~$n$~$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}c=0$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct=c$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=2ct$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=3ct^2$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^n=nct^{n-1}$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}e^t=e^t$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}sin(t)=cos(t)$~$$ |
---|

Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}cos(t)=-sin(t)$~$$ |
---|

Diagonal lemma | $~$T$~$ |
---|

Diagonal lemma | $~$S$~$ |
---|

Diagonal lemma | $~$T\vdash S\iff F(\ulcorner S \urcorner)$~$ |
---|

Diagonal lemma | $~$\phi(x)$~$ |
---|

Diagonal lemma | $~$T$~$ |
---|

Diagonal lemma | $~$\phi(x)$~$ |
---|

Diagonal lemma | $~$x$~$ |
---|

Diagonal lemma | $~$S$~$ |
---|

Diagonal lemma | $~$T\vdash S\leftrightarrow \phi(\ulcorner S\urcorner)$~$ |
---|

Diagonal lemma | $~$\neg \square_{PA} (x)$~$ |
---|

Diagonal lemma | $~$PA$~$ |
---|

Diagonal lemma | $~$x$~$ |
---|

Diagonal lemma | $~$G$~$ |
---|

Diagonal lemma | $~$PA\vdash G\leftrightarrow \neg \square_{PA} (\ulcorner G\urcorner)$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n}
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\vec{y_n}$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^{th}$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n \times 1$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^th$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\mathbf{W_n}$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_{n-1} \times l_{n}$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n-1$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\vec{b_n}$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^th$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$(n-1)^th$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n\times1$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$w$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
f(x)=w\times x
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$f(x)=m\times x$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$y=mx+b$~$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=\mathbf{W_n}^T \times \vec{y_{n-1}} + 1 \times \vec{b_n}
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=
\left[ \begin{array}{c}
x, \\ 1
\end{array} \right]^T
\cdot
\left[ \begin{array}{c}
\mathbf{W_n},
\\ \vec{b_n}
\end{array} \right]
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}} = \vec{y_{new_{n-1}}}^T \times \vec{W_{new}}
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{W_{new}} =\vec{W_{new}}-\frac{\delta W_{new}}{\delta Error}
$~$$ |
---|

Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{W_{new}} =Activation(\vec{W_{new}}-\frac{\delta W_{new}}{\delta Error})
$~$$ |
---|

Dihedral group | $~$D_{2n}$~$ |
---|

Dihedral group | $~$n$~$ |
---|

Dihedral group | $$~$D_{2n} \cong \langle a, b \mid a^n, b^2, b a b^{-1} = a^{-1} \rangle$~$$ |
---|

Dihedral group | $~$a$~$ |
---|

Dihedral group | $~$b$~$ |
---|

Dihedral group | $~$D_{2n}$~$ |
---|

Dihedral group | $~$n > 2$~$ |
---|

Dihedral group | $~$D_{2n}$~$ |
---|

Dihedral group | $~$S_n$~$ |
---|

Dihedral group | $~$a = (123 \dots n)$~$ |
---|

Dihedral group | $~$b = (2, n)(3, n-1) \dots (\frac{n}{2}+1, \frac{n}{2}+3)$~$ |
---|

Dihedral group | $~$n$~$ |
---|

Dihedral group | $~$b = (2, n)(3, n-1)\dots(\frac{n-1}{2}, \frac{n+1}{2})$~$ |
---|

Dihedral group | $~$n$~$ |
---|

Dihedral group | $~$D_6$~$ |
---|

Dihedral group | $~$\langle a, b \mid b^2, b a b^{-1} = a^{-1} \rangle$~$ |
---|

Dihedral group | $~$D_{2n}$~$ |
---|

Dihedral group | $~$\mathbb{R}^2$~$ |
---|

Dihedral group | $~$x=0$~$ |
---|

Dihedral group | $~$D_{2n}$~$ |
---|

Dihedral groups are non-abelian | $~$n \geq 3$~$ |
---|

Dihedral groups are non-abelian | $~$n$~$ |
---|

Dihedral groups are non-abelian | $~$D_{2n}$~$ |
---|

Dihedral groups are non-abelian | $~$\langle a, b \mid a^n, b^2, bab^{-1} = a^{-1} \rangle$~$ |
---|

Dihedral groups are non-abelian | $~$ba = a^{-1} b = a^{-2} a b$~$ |
---|

Dihedral groups are non-abelian | $~$ab = ba$~$ |
---|

Dihedral groups are non-abelian | $~$a^2$~$ |
---|

Dihedral groups are non-abelian | $~$a$~$ |
---|

Dihedral groups are non-abelian | $~$n > 2$~$ |
---|

Dihedral groups are non-abelian | $~$ab$~$ |
---|

Dihedral groups are non-abelian | $~$ba$~$ |
---|

Direct sum of vector spaces | $~$U$~$ |
---|

Direct sum of vector spaces | $~$W,$~$ |
---|

Direct sum of vector spaces | $~$U \oplus W,$~$ |
---|

Direct sum of vector spaces | $~$U$~$ |
---|

Direct sum of vector spaces | $~$W,$~$ |
---|

Direct sum of vector spaces | $~$U$~$ |
---|

Direct sum of vector spaces | $~$W$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$S_n$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$a_i, b_j$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$S_n$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$\sigma$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$\tau$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$\sigma(a_i) = (b_1 b_2 \dots b_m)[(a_1 a_2 \dots a_k)(a_i)] = (b_1 b_2 \dots b_m)(a_{i+1}) = a_{i+1}$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$a_{k+1}$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$a_1$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$\tau(a_i) = (a_1 a_2 \dots a_k)[(b_1 b_2 \dots b_m)(a_i)] = (a_1 a_2 \dots a_k)(a_i) = a_{i+1}$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$a_i$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$b_j$~$ |
---|

Disjoint cycles commute in symmetric groups | $~$\{1,2,\dots, n\}$~$ |
---|

Disjoint union of sets | $~$\sqcup$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|

Disjoint union of sets | $~$B = \{8, 9\}$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$\{6,7,8,9\}$~$ |
---|

Disjoint union of sets | $~$A \sqcup B = \{6,7,8,9\}$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A \cup B$~$ |
---|

Disjoint union of sets | $~$\sqcup$~$ |
---|

Disjoint union of sets | $~$\cup$~$ |
---|

Disjoint union of sets | $~$\{1,2\} \sqcup \{1,3\} = \{1,2,3\}$~$ |
---|

Disjoint union of sets | $~$\{1,2\} \cup \{1,3\} = \{1,2,3\}$~$ |
---|

Disjoint union of sets | $~$1$~$ |
---|

Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|

Disjoint union of sets | $~$B = \{6,8\}$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$a$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$1$~$ |
---|

Disjoint union of sets | $~$(a, 1)$~$ |
---|

Disjoint union of sets | $~$a$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$A'$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $$~$A' = \{ (a, 1) : a \in A \}$~$$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$1$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$2$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$B'$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $$~$B' = \{ (b,2) : b \in B \}$~$$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$A'$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$A'$~$ |
---|

Disjoint union of sets | $~$a \mapsto (a,1)$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$B'$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A'$~$ |
---|

Disjoint union of sets | $~$B'$~$ |
---|

Disjoint union of sets | $~$A'$~$ |
---|

Disjoint union of sets | $~$1$~$ |
---|

Disjoint union of sets | $~$B'$~$ |
---|

Disjoint union of sets | $~$2$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A' \sqcup B'$~$ |
---|

Disjoint union of sets | $~$\sqcup$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|

Disjoint union of sets | $~$B=\{6,8\}$~$ |
---|

Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|

Disjoint union of sets | $~$B=\{6,8\}$~$ |
---|

Disjoint union of sets | $~$\sqcup$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$6$~$ |
---|

Disjoint union of sets | $~$A' = \{ (6, 1), (7, 1) \}$~$ |
---|

Disjoint union of sets | $~$B' = \{ (6, 2), (8, 2) \}$~$ |
---|

Disjoint union of sets | $$~$A \sqcup B = \{ (6,1), (7,1), (6,2), (8,2) \}$~$$ |
---|

Disjoint union of sets | $~$A \cup B = \{ 6, 7, 8 \}$~$ |
---|

Disjoint union of sets | $~$6$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$6$~$ |
---|

Disjoint union of sets | $~$(6,1)$~$ |
---|

Disjoint union of sets | $~$(6,2)$~$ |
---|

Disjoint union of sets | $~$A = \{1,2\}$~$ |
---|

Disjoint union of sets | $~$B = \{3,4\}$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$A \cup B = \{1,2,3,4\}$~$ |
---|

Disjoint union of sets | $~$A' \cup B' = \{(1,1), (2,1), (3,2), (4,2) \}$~$ |
---|

Disjoint union of sets | $~$A' = \{(1,1), (2,1)\}$~$ |
---|

Disjoint union of sets | $~$B' = \{(3,2), (4,2) \}$~$ |
---|

Disjoint union of sets | $~$A = B = \{6,7\}$~$ |
---|

Disjoint union of sets | $~$A' = \{(6,1), (7,1)\}$~$ |
---|

Disjoint union of sets | $~$B' = \{(6,2), (7,2)\}$~$ |
---|

Disjoint union of sets | $$~$A \sqcup B = \{(6,1),(7,1),(6,2),(7,2)\}$~$$ |
---|

Disjoint union of sets | $~$A = \mathbb{N}$~$ |
---|

Disjoint union of sets | $~$B = \{ 1, 2, x \}$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$\mathbb{N}$~$ |
---|

Disjoint union of sets | $~$0$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$\{1,2,x\}$~$ |
---|

Disjoint union of sets | $~$x$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$A' = \{ (0,1), (1,1), (2,1), (3,1), \dots\}$~$ |
---|

Disjoint union of sets | $~$B' = \{(1,2), (2,2), (x,2)\}$~$ |
---|

Disjoint union of sets | $$~$\{(0,1), (1,1),(2,1),(3,1), \dots, (1,2),(2,2),(x,2)\}$~$$ |
---|

Disjoint union of sets | $~$A = \mathbb{N}$~$ |
---|

Disjoint union of sets | $~$B = \{x, y\}$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$\{ 0,1,2,\dots, x, y \}$~$ |
---|

Disjoint union of sets | $~$\{(0,1), (1,1), (2,1), \dots, (x,2), (y,2)\}$~$ |
---|

Disjoint union of sets | $~$A \sqcup B \sqcup C$~$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$A \cup B \cup C$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$C$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$B$~$ |
---|

Disjoint union of sets | $~$C$~$ |
---|

Disjoint union of sets | $~$A$~$ |
---|

Disjoint union of sets | $~$C$~$ |
---|

Disjoint union of sets | $~$A' = \{(a, 1) : a \in A \}$~$ |
---|

Disjoint union of sets | $~$B' = \{ (b, 2) : b \in B \}$~$ |
---|

Disjoint union of sets | $~$C' = \{ (c, 3) : c \in C \}$~$ |
---|

Disjoint union of sets | $~$A \sqcup B \sqcup C$~$ |
---|

Disjoint union of sets | $~$A' \cup B' \cup C'$~$ |
---|

Disjoint union of sets | $$~$\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} A_i$~$$ |
---|

Disjoint union of sets | $~$A_i$~$ |
---|

Disjoint union of sets | $$~$\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} A'_i$~$$ |
---|

Disjoint union of sets | $~$A'_i = \{ (a, i) : a \in A_i \}$~$ |
---|

Disjoint union of sets | $$~$\bigsqcup_{n \in \mathbb{N}} \{0, 1,2,\dots,n\} = \{(0,0)\} \cup \{(0,1), (1,1) \} \cup \{ (0,2), (1,2), (2,2)\} \cup \dots = \{ (n, m) : n \leq m \}$~$$ |
---|

Disjoint union of sets | $~$A \sqcup B$~$ |
---|

Disjoint union of sets | $~$A' \cup B'$~$ |
---|

Disjoint union of sets | $~$A' = \{ (a, 2) : a \in A \}$~$ |
---|

Disjoint union of sets | $~$B' = \{ (b,1) : b \in B \}$~$ |
---|

Division of rational numbers (Math 0) | $~$1$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{4}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$1$~$ |
---|

Division of rational numbers (Math 0) | $$~$1 + \frac{1}{3} = \frac{1}{1} + \frac{1}{3} = \frac{3 \times 1 + 1 \times 1}{3 \times 1} = \frac{3+1}{3} = \frac{4}{3}$~$$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{4}{3}$~$ |
---|

Division of rational numbers (Math 0) | $~$x$~$ |
---|

Division of rational numbers (Math 0) | $~$y$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{x}{y}$~$ |
---|

Division of rational numbers (Math 0) | $~$x$~$ |
---|

Division of rational numbers (Math 0) | $~$y$~$ |
---|

Division of rational numbers (Math 0) | $~$a/n$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$1$~$ |
---|

Division of rational numbers (Math 0) | $~$m$~$ |
---|

Division of rational numbers (Math 0) | $~$1$~$ |
---|

Division of rational numbers (Math 0) | $~$m$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$n$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$n$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$a$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$n$~$ |
---|

Division of rational numbers (Math 0) | $~$a$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$n$~$ |
---|

Division of rational numbers (Math 0) | $~$n$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m} \times \frac{1}{n}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|

Division of rational numbers (Math 0) | $$~$\frac{a}{m} / n = \frac{a}{m \times n}$~$$ |
---|

Division of rational numbers (Math 0) | $~$x$~$ |
---|

Division of rational numbers (Math 0) | $~$x$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{-1}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{1} = 1$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{1}{1} = 1$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{-1}{-1}$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{-1}{-1}$~$ |
---|

Division of rational numbers (Math 0) | $~$1$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{-1}{-1} = 1$~$ |
---|

Division of rational numbers (Math 0) | $~$\frac{a}{m} \times \frac{b}{n} = \frac{a \times b}{m \times n}$~$ |
---|

Division of rational numbers (Math 0) | $$~$\frac{1}{-m} = \frac{1}{-m} \times 1 = \frac{1}{-m} \times \frac{-1}{-1} = \frac{-1 \times 1}{-m \times -1} = \frac{-1}{m}$~$$ |
---|

Division of rational numbers (Math 0) | $~$\frac{a}{-b} = \frac{-a}{b}$~$ |
---|

Domain (of a function) | $~$\operatorname{dom}(f)$~$ |
---|

Domain (of a function) | $~$f : X \to Y$~$ |
---|

Domain (of a function) | $~$X$~$ |
---|

Domain (of a function) | $~$+$~$ |
---|

Domain (of a function) | $~$(x, y)$~$ |
---|

Domain (of a function) | $~$y$~$ |
---|

Effective number of political parties | $~$1, 2, \ldots, n$~$ |
---|

Effective number of political parties | $~$p_n$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$0$~$ |
---|

Effective number of political parties | $~$1$~$ |
---|

Effective number of political parties | $~$\displaystyle \frac{1}{\sum_{i=1}^n p_i^2}$~$ |
---|

Effective number of political parties | $~$x$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$k$~$ |
---|

Effective number of political parties | $~$k$~$ |
---|

Effective number of political parties | $~$k = 1$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$p_i$~$ |
---|

Effective number of political parties | $~$n$~$ |
---|

Effective number of political parties | $~$1/n$~$ |
---|

Effective number of political parties | $~$p_i$~$ |
---|

Effective number of political parties | $~$p_i$~$ |
---|

Effective number of political parties | $~$p_i$~$ |
---|

Effective number of political parties | $~$(p_1 \cdot p_1) + (p_2 \cdot p_2) + \ldots + (p_n \cdot p_n) = \sum_{i=1}^n p_i^2$~$ |
---|

Eigenvalues and eigenvectors | $~$A$~$ |
---|

Eigenvalues and eigenvectors | $~$v$~$ |
---|

Eigenvalues and eigenvectors | $~$Av = \lambda v$~$ |
---|

Eigenvalues and eigenvectors | $~$v$~$ |
---|

Eigenvalues and eigenvectors | $~$A$~$ |
---|

Eigenvalues and eigenvectors | $~$\lambda$~$ |
---|

Eigenvalues and eigenvectors | $~$A$~$ |
---|

Eigenvalues and eigenvectors | $~$v$~$ |
---|

Eigenvalues and eigenvectors | $~$|\lambda| > 1$~$ |
---|

Eigenvalues and eigenvectors | $~$|\lambda| < 1$~$ |
---|

Eigenvalues and eigenvectors | $~$\lambda < 0$~$ |
---|

Elementary Algebra | $$~$2 + 2 = 4$~$$ |
---|

Elementary Algebra | $~$2 < 4$~$ |
---|

Elementary Algebra | $~$5 > 1$~$ |
---|

Elementary Algebra | $$~$2 + (3 \times 4) = 14$~$$ |
---|

Elementary Algebra | $$~$(2 + 3) \times 4 = 20$~$$ |
---|

Elementary Algebra | $~$2 + 3 \times 4$~$ |
---|

Elementary Algebra | $~$2 + (3 \times 4)$~$ |
---|

Elementary Algebra | $~$2+2=4$~$ |
---|

Elementary Algebra | $~$(2 + 2) + 3 = 4 + 3$~$ |
---|

Elementary Algebra | $~$2^3 \times 2^4$~$ |
---|

Elementary Algebra | $~$2^3 = 2 \times 2 \times 2$~$ |
---|

Elementary Algebra | $~$2 \times 2 \times 2 = 8$~$ |
---|

Elementary Algebra | $~$2^3 = 8$~$ |
---|

Elementary Algebra | $~$2^4 = 2 \times 2 \times 2 \times 2 = 16$~$ |
---|

Elementary Algebra | $~$2^3 \times 2^4 = 8 \times 16$~$ |
---|

Elementary Algebra | $~$2^3\times 2^4 = 128$~$ |
---|

Elementary Algebra | $~$0 \times 3 = 0$~$ |
---|

Elementary Algebra | $~$0 \times -4 = 0$~$ |
---|

Elementary Algebra | $~$0 \times 1224 = 0$~$ |
---|

Elementary Algebra | $~$0 \times \text{any number} = 0$~$ |
---|

Elementary Algebra | $~$0 \times x = 0$~$ |
---|

Elementary Algebra | $~$x$~$ |
---|

Elementary Algebra | $$~$ a + b = b + a$~$$ |
---|

Elementary Algebra | $$~$ a \times b = b\times a$~$$ |
---|

Elementary Algebra | $$~$ 0 + a = a$~$$ |
---|

Elementary Algebra | $$~$ 1 \times a = a$~$$ |
---|

Elementary Algebra | $$~$ (a + b) + c = a + (b + c)$~$$ |
---|

Elementary Algebra | $$~$ (a \times b ) \times c = a \times (b\times c)$~$$ |
---|

Elementary Algebra | $$~$ a \times (b + c) = a\times b + a\times c$~$$ |
---|

Elementary Algebra | $$~$ a + (-a) = a - a = 0 $~$$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1,$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|

Empirical probabilities are not exactly 0 or 1 | $~$1,$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$x$~$ |
---|

Empty set | $~$x$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$x$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$x$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$X$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$B$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$B$~$ |
---|

Empty set | $~$B$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$A = B$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\{ \emptyset \}$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\{\emptyset\}$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$P$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$P$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $$~$\exists B \forall x : x∉B$~$$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$A$~$ |
---|

Empty set | $~$B$~$ |
---|

Empty set | $~$\forall x : x∉A$~$ |
---|

Empty set | $~$\forall x: x∉B$~$ |
---|

Empty set | $~$\forall x : (x ∈ A \Leftrightarrow x ∈ B)$~$ |
---|

Empty set | $~$A=B$~$ |
---|

Empty set | $~$x$~$ |
---|

Empty set | $~$(x ∈ A \Leftrightarrow x ∈ B)$~$ |
---|

Empty set | $~$x \not \in A$~$ |
---|

Empty set | $~$x \not \in B$~$ |
---|

Empty set | $~$\phi$~$ |
---|

Empty set | $~$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \phi(x))$~$ |
---|

Empty set | $~$\phi$~$ |
---|

Empty set | $~$\bot$~$ |
---|

Empty set | $~$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \bot)$~$ |
---|

Empty set | $~$x \in b \Leftrightarrow (x \in a \wedge \bot)$~$ |
---|

Empty set | $~$x \in b \Leftrightarrow \bot$~$ |
---|

Empty set | $~$x \notin b$~$ |
---|

Empty set | $~$\forall a \exists b \forall x : x \notin b$~$ |
---|

Empty set | $~$a$~$ |
---|

Empty set | $~$\{\emptyset\}$~$ |
---|

Empty set | $~$\{\emptyset\} \not= \emptyset$~$ |
---|

Empty set | $~$\emptyset ∈ \{\emptyset\}$~$ |
---|

Empty set | $~$\emptyset ∉ \emptyset$~$ |
---|

Empty set | $~$\{\emptyset\}$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$|\{\emptyset\}| = 1$~$ |
---|

Empty set | $~$\emptyset$~$ |
---|

Empty set | $~$|\emptyset| = 0$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$m, n \in$~$ |
---|

Emulating digits | $~$\mathbb N$~$ |
---|

Emulating digits | $~$m < n,$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$7$~$ |
---|

Emulating digits | $~$m > n,$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$n^2$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$(x, y)$~$ |
---|

Emulating digits | $~$0 \le x < n$~$ |
---|

Emulating digits | $~$0 \le y < n$~$ |
---|

Emulating digits | $~$(x, y)$~$ |
---|

Emulating digits | $~$xn + y.$~$ |
---|

Emulating digits | $~$x = y = 0$~$ |
---|

Emulating digits | $~$n^2 - 1$~$ |
---|

Emulating digits | $~$x = y = n-1$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$n^2$~$ |
---|

Emulating digits | $~$n^3$~$ |
---|

Emulating digits | $~$(x, y, z)$~$ |
---|

Emulating digits | $~$xn^2 + yn + z$~$ |
---|

Emulating digits | $~$n^4$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$a$~$ |
---|

Emulating digits | $~$n^a > m,$~$ |
---|

Emulating digits | $~$a$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$n$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Emulating digits | $~$m$~$ |
---|

Encoding trits with GalCom bits | $~$\log_2(3) \approx 1.585$~$ |
---|

Encoding trits with GalCom bits | $~$2 - \frac{1}{3} \approx 1.67$~$ |
---|

Environmental goals | $~$E_{1,t} \ldots E_{n,t}$~$ |
---|

Environmental goals | $~$t.$~$ |
---|

Environmental goals | $~$S_t$~$ |
---|

Environmental goals | $~$E_t$~$ |
---|

Environmental goals | $~$A_t$~$ |
---|

Environmental goals | $~$t.$~$ |
---|

Environmental goals | $~$R_t$~$ |
---|

Environmental goals | $~$E_t$~$ |
---|

Environmental goals | $~$A_t$~$ |
---|

Environmental goals | $~$E_{t+1}$~$ |
---|

Environmental goals | $~$E_t$~$ |
---|

Environmental goals | $~$A_t.$~$ |
---|

Environmental goals | $~$E_{1,t}$~$ |
---|

Environmental goals | $~$t.$~$ |
---|

Environmental goals | $~$A_t$~$ |
---|

Environmental goals | $~$\theta.$~$ |
---|

Environmental goals | $~$A_t$~$ |
---|

Environmental goals | $~$\theta$~$ |
---|

Environmental goals | $~$E_1 \ldots E_m$~$ |
---|

Environmental goals | $~$E_{m+1} \ldots E_n$~$ |
---|

Environmental goals | $~$E_{m+1} \ldots E_n$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$E_1 \ldots E_m$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_{m+1, t} \ldots E_{n,t} = 0 \implies R_t=E_{1, t}.$~$ |
---|

Environmental goals | $~$E_{m+1} \ldots E_n,$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$A_t$~$ |
---|

Environmental goals | $~$S_{t+1}$~$ |
---|

Environmental goals | $~$S_{1, t}$~$ |
---|

Environmental goals | $~$E_{1, t},$~$ |
---|

Environmental goals | $~$S_1.$~$ |
---|

Environmental goals | $~$S_1$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$Q$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$S_1$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Environmental goals | $~$S_1,$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_1,$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$Q$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$Q.$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Environmental goals | $~$Q$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$Q.$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$R$~$ |
---|

Environmental goals | $~$R.$~$ |
---|

Environmental goals | $~$E_1$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Environmental goals | $~$S$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Environmental goals | $~$Q$~$ |
---|

Environmental goals | $~$S$~$ |
---|

Environmental goals | $~$E_1.$~$ |
---|

Equaliser (category theory) | $~$f, g: A \to B$~$ |
---|

Equaliser (category theory) | $~$E$~$ |
---|

Equaliser (category theory) | $~$e: E \to A$~$ |
---|

Equaliser (category theory) | $~$ge = fe$~$ |
---|

Equaliser (category theory) | $~$ge = fe$~$ |
---|

Equaliser (category theory) | $~$X$~$ |
---|

Equaliser (category theory) | $~$x: X \to A$~$ |
---|

Equaliser (category theory) | $~$fx = gx$~$ |
---|

Equaliser (category theory) | $~$\bar{x} : X \to A$~$ |
---|

Equaliser (category theory) | $~$e \bar{x} = x$~$ |
---|

Equivalence relation | $~$\sim$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$x \in S$~$ |
---|

Equivalence relation | $~$x \sim x$~$ |
---|

Equivalence relation | $~$x,y \in S$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$y \sim x$~$ |
---|

Equivalence relation | $~$x,y,z \in S$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$y \sim z$~$ |
---|

Equivalence relation | $~$x \sim z$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$\sim$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$x \in S$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$x$~$ |
---|

Equivalence relation | $~$[x]=\{y \in S \mid x \sim y\}$~$ |
---|

Equivalence relation | $~$x$~$ |
---|

Equivalence relation | $~$[x]$~$ |
---|

Equivalence relation | $~$S/\sim = \{[x] \mid x \in S\}$~$ |
---|

Equivalence relation | $~$x \in [x]$~$ |
---|

Equivalence relation | $~$[x]=[y]$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$S$~$ |
---|

Equivalence relation | $~$A$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$U \in A$~$ |
---|

Equivalence relation | $~$x,y \in U$~$ |
---|

Equivalence relation | $~$[x] \in A$~$ |
---|

Equivalence relation | $~$A=S/\sim$~$ |
---|

Equivalence relation | $~$f: S \to U$~$ |
---|

Equivalence relation | $~$f^*: S/\sim \to U$~$ |
---|

Equivalence relation | $~$U$~$ |
---|

Equivalence relation | $~$f^*([x])$~$ |
---|

Equivalence relation | $~$f(x)$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$f(x) \neq f(y)$~$ |
---|

Equivalence relation | $~$f^*([x])=f^*([y])$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$f(x)=f(y)$~$ |
---|

Equivalence relation | $~$f: S \to S$~$ |
---|

Equivalence relation | $~$f^*: S/\sim \to S/\sim$~$ |
---|

Equivalence relation | $~$f^*([x])=[f(x)]$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$[f(x)]=[f(y)]$~$ |
---|

Equivalence relation | $~$f(x) \sim f(y)$~$ |
---|

Equivalence relation | $~$x \sim y$~$ |
---|

Equivalence relation | $~$n|x-y$~$ |
---|

Equivalence relation | $~$n \in \mathbb N$~$ |
---|

Equivalence relation | $~$n$~$ |
---|

Equivalence relation | $~$n$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$1$~$ |
---|

Euclid's Lemma on prime numbers | $~$x, y$~$ |
---|

Euclid's Lemma on prime numbers | $~$ax+py = 1$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c$~$ |
---|

Euclid's Lemma on prime numbers | $~$c \mid a$~$ |
---|

Euclid's Lemma on prime numbers | $~$c \mid p$~$ |
---|

Euclid's Lemma on prime numbers | $~$d$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$d \mid c$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c \mid p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c = p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c=1$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$c \mid a$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$a$~$ |
---|

Euclid's Lemma on prime numbers | $~$c = 1$~$ |
---|

Euclid's Lemma on prime numbers | $~$b$~$ |
---|

Euclid's Lemma on prime numbers | $~$abx + pby = b$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$ab$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|

Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|

Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|

Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|

Euclid's Lemma on prime numbers | $~$pq$~$ |
---|

Euclid's Lemma on prime numbers | $~$p, q$~$ |
---|

Euclid's Lemma on prime numbers | $~$1$~$ |
---|

Euclid's Lemma on prime numbers | $~$pq$~$ |
---|

Euclid's Lemma on prime numbers | $~$p$~$ |
---|

Euclid's Lemma on prime numbers | $~$q$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$n > 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$n$~$ |
---|

Euclidean domains are principal ideal domains | $~$n < 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$-n$~$ |
---|

Euclidean domains are principal ideal domains | $~$R$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi: \mathbb{R} \setminus \{ 0 \} \to \mathbb{N}^{\geq 0}$~$ |
---|

Euclidean domains are principal ideal domains | $~$a$~$ |
---|

Euclidean domains are principal ideal domains | $~$b$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi(a) \leq \phi(b)$~$ |
---|

Euclidean domains are principal ideal domains | $~$a$~$ |
---|

Euclidean domains are principal ideal domains | $~$b$~$ |
---|

Euclidean domains are principal ideal domains | $~$a$~$ |
---|

Euclidean domains are principal ideal domains | $~$q$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$a = qb+r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi(r) < \phi(b)$~$ |
---|

Euclidean domains are principal ideal domains | $~$I \subseteq R$~$ |
---|

Euclidean domains are principal ideal domains | $~$I$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha: R \to S$~$ |
---|

Euclidean domains are principal ideal domains | $~$r \in R$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(x) = 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$x$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$r = 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$x$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$ar$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(ar) = \alpha(a) \alpha(r) = \alpha(a) \times 0 = 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$x$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$x = ar+b$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi(b) < \phi(r)$~$ |
---|

Euclidean domains are principal ideal domains | $~$b$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(x) = \alpha(ar)+\alpha(b)$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(r) = 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(x) = \alpha(b)$~$ |
---|

Euclidean domains are principal ideal domains | $~$b$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(b)$~$ |
---|

Euclidean domains are principal ideal domains | $~$0$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(x)$~$ |
---|

Euclidean domains are principal ideal domains | $~$\alpha(x) = 0$~$ |
---|

Euclidean domains are principal ideal domains | $~$x$~$ |
---|

Euclidean domains are principal ideal domains | $~$r$~$ |
---|

Euclidean domains are principal ideal domains | $~$\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]$~$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$F(X)$~$ |
---|

Every group is a quotient of a free group | $~$X$~$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$F(X)$~$ |
---|

Every group is a quotient of a free group | $~$T: \mathcal{C} \to \mathcal{C}$~$ |
---|

Every group is a quotient of a free group | $~$\mathcal{C}$~$ |
---|

Every group is a quotient of a free group | $~$(A, \alpha)$~$ |
---|

Every group is a quotient of a free group | $~$T$~$ |
---|

Every group is a quotient of a free group | $~$\alpha: TA \to A$~$ |
---|

Every group is a quotient of a free group | $~$F(G)$~$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$\theta: F(G) \to G$~$ |
---|

Every group is a quotient of a free group | $~$(a_1, a_2, \dots, a_n)$~$ |
---|

Every group is a quotient of a free group | $~$a_1 a_2 \dots a_n$~$ |
---|

Every group is a quotient of a free group | $~$F(G)$~$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$w_1 = (a_1, \dots, a_m)$~$ |
---|

Every group is a quotient of a free group | $~$w_2 = (b_1, \dots, b_n)$~$ |
---|

Every group is a quotient of a free group | $$~$\theta(w_1 w_2) = \theta(a_1, \dots, a_m, b_1, \dots, b_m) = a_1 \dots a_m b_1 \dots b_m = \theta(w_1) \theta(w_2)$~$$ |
---|

Every group is a quotient of a free group | $~$G$~$ |
---|

Every group is a quotient of a free group | $~$F(G)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$S_n$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\tau_1, \dots, \tau_k$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma = \tau_k \tau_{k-1} \dots \tau_1$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(123)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(23)(13)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$3$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_1 a_2 \dots a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{r-1} a_r) (a_{r-2} a_r) \dots (a_2 a_r) (a_1 a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_1 a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_2 a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i-1} a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_i a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_r$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i+1} a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_r$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i+2} a_r), \dots, (a_{r-1} a_r)$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|

Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|

Examination through isomorphism | $~$(X,d)$~$ |
---|

Examination through isomorphism | $~$d(x,y)$~$ |
---|

Examination through isomorphism | $~$x,y \in X$~$ |
---|

Examination through isomorphism | $~$[0,1]$~$ |
---|

Examination through isomorphism | $~$[0,2]$~$ |
---|

Examination through isomorphism | $~$\mathbb{R}$~$ |
---|

Examination through isomorphism | $~$\mathbb{R}$~$ |
---|

Examination through isomorphism | $~$f : [0,1] \to [0,2]$~$ |
---|

Examination through isomorphism | $~$g : [0,2] \to [0,1]$~$ |
---|

Examination through isomorphism | $~$fg$~$ |
---|

Examination through isomorphism | $~$gf$~$ |
---|

Examination through isomorphism | $~$f$~$ |
---|

Examination through isomorphism | $~$2$~$ |
---|

Examination through isomorphism | $~$g$~$ |
---|

Examination through isomorphism | $~$2$~$ |
---|

Examination through isomorphism | $~$[0,1]$~$ |
---|

Examination through isomorphism | $~$1$~$ |
---|

Examination through isomorphism | $~$[0,2]$~$ |
---|

Examination through isomorphism | $~$2$~$ |
---|

Examination through isomorphism | $~$\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|

Examination through isomorphism | $~$A \times (B \times C)$~$ |
---|

Examination through isomorphism | $~$(a,(b,c))$~$ |
---|

Examination through isomorphism | $~$(A \times B) \times C$~$ |
---|

Examination through isomorphism | $~$((a,b),c)$~$ |
---|

Examination through isomorphism | $~$\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|

Examination through isomorphism | $~$(A,B,C) \mapsto A \times (B \times C)$~$ |
---|

Examination through isomorphism | $~$(A,B,C) \mapsto (A \times B) \times C$~$ |
---|

Examination through isomorphism | $~$\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|

Example: Dragon Pox | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|

Example: Dragon Pox | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|

Example: Dragon Pox | $~$\bP(D) = 0.4$~$ |
---|

Example: Dragon Pox | $~$\bP(S \mid D) = 0.7$~$ |
---|

Example: Dragon Pox | $~$\bP(S \mid \neg D) = 0.2$~$ |
---|

Example: Dragon Pox | $~$(C)$~$ |
---|

Example: Dragon Pox | $~$(\neg C)$~$ |
---|

Example: Dragon Pox | $~$(L)$~$ |
---|

Example: Dragon Pox | $~$(\neg L)$~$ |
---|

Example: Dragon Pox | $$~$
\begin{align}
\bP(L \mid \;\;D,\;\;C) &= 0.4\\
\bP(L \mid \;\;D,\neg C) &= 0.1\\
\bP(L \mid \neg D,\;\;C) &= 0.7\\
\bP(L \mid \neg D,\neg C) &= 0.9
\end{align}
$~$$ |
---|

Example: Dragon Pox | $~$D$~$ |
---|

Example: Dragon Pox | $~$\bP(D) = 0.4$~$ |
---|

Example: Dragon Pox | $~$S$~$ |
---|

Example: Dragon Pox | $~$\bP(S \mid D) = 0.7$~$ |
---|

Example: Dragon Pox | $~$\bP(S \mid \neg D) = 0.2$~$ |
---|

Example: Dragon Pox | $~$(C)$~$ |
---|

Example: Dragon Pox | $~$(\neg C)$~$ |
---|

Example: Dragon Pox | $~$(L)$~$ |
---|

Example: Dragon Pox | $~$D$~$ |
---|

Example: Dragon Pox | $~$C$~$ |
---|

Example: Dragon Pox | $$~$
\begin{align}
\bP(L \mid \;\;D,\;\;C) &= 0.4\\
\bP(L \mid \;\;D,\neg C) &= 0.1\\
\bP(L \mid \neg D,\;\;C) &= 0.7\\
\bP(L \mid \neg D,\neg C) &= 0.9
\end{align}
$~$$ |
---|

Example: Dragon Pox | $~$\bP(L \mid D,C) > \bP(L \mid D,\neg C)$~$ |
---|

Example: Dragon Pox | $~$\neg D$~$ |
---|

Example: Dragon Pox | $~$\bP(L \mid \neg D,C) < \bP(L \mid \neg D,\neg C)$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$\log_b(n).$~$ |
---|

Exchange rates between digits | $~$2^\text{3,000,000,000,000}$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$2^n$~$ |
---|

Exchange rates between digits | $~$2^4=16$~$ |
---|

Exchange rates between digits | $~$2^6 < 101 < 2^7$~$ |
---|

Exchange rates between digits | $~$2^{12} < 8000 < 2^{13}$~$ |
---|

Exchange rates between digits | $~$2^{13} < 15,000 < 2^{14}$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$3n$~$ |
---|

Exchange rates between digits | $~$10^n > 2^{3n}$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$3n$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$10^n$~$ |
---|

Exchange rates between digits | $~$2^3$~$ |
---|

Exchange rates between digits | $~$2^{3n}$~$ |
---|

Exchange rates between digits | $~$n$~$ |
---|

Exchange rates between digits | $~$2^{3(n-1)}$~$ |
---|

Exchange rates between digits | $~$n \ge 11,$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$10^{10} < 2^{35}.$~$ |
---|

Exchange rates between digits | $~$2^{33} < 10^{10} < 2^{34},$~$ |
---|

Exchange rates between digits | $~$2^{332} < 10^{100} < 2^{333},$~$ |
---|

Exchange rates between digits | $~$p$~$ |
---|

Exchange rates between digits | $~$2^p > 10$~$ |
---|

Exchange rates between digits | $~$2^p < 10$~$ |
---|

Exchange rates between digits | $~$p$~$ |
---|

Exchange rates between digits | $~$2^p = 10,$~$ |
---|

Exchange rates between digits | $~$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$~$ |
---|

Exchange rates between digits | $~$2 + 2 + 2 + 2 + 2 = 10.$~$ |
---|

Exchange rates between digits | $~$p$~$ |
---|

Exchange rates between digits | $~$p$~$ |
---|

Exchange rates between digits | $~$2^p = 10$~$ |
---|

Exchange rates between digits | $~$\log_2(10),$~$ |
---|

Exchange rates between digits | $~$\log_b(x)$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$\log_2(6) \approx 2.58$~$ |
---|

Exchange rates between digits | $~$2^2 < 6 < 2^3$~$ |
---|

Exchange rates between digits | $~$2^{25} < 6^{10} < 2^{26}$~$ |
---|

Exchange rates between digits | $~$2^{258} < 6^{100} < 2^{259}.$~$ |
---|

Exchange rates between digits | $~$\log_2(6)$~$ |
---|

Exchange rates between digits | $~$\log_b(x)$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$\log_b(x)$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$\log_x(b) = \frac{1}{\log_b(x)}$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$b$~$ |
---|

Exchange rates between digits | $~$x$~$ |
---|

Exchange rates between digits | $~$\log_{1.5}(2.5)$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{\mathbb{P}}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{T}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{\mathbb{P}}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{LIA}$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
---|

Existence Proof of Logical Inductor | $~$-b$~$ |
---|

Existence Proof of Logical Inductor | $~$-b$~$ |
---|

Existence Proof of Logical Inductor | $~$\overline{T}$~$ |
---|

Existence Proof of Logical Inductor | $~$n$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathbb{P}_n$~$ |
---|

Existence Proof of Logical Inductor | $~$T_n(\mathbb{P}_{\leq n})$~$ |
---|

Existence Proof of Logical Inductor | $~$\text{fix}(\mathbb{V})(\phi) := \max{(0,\min{(1, \mathbb{V}(\phi) + T(\mathbb{P}_{\leq n-1},\mathbb{V})[\phi])})}$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathbb{V}^{\text{fix}}$~$ |
---|

Existence Proof of Logical Inductor | $~$\phi$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathbb{V}^{\text{fix}}(\phi)= \max{(0,\min{(1, \mathbb{V}^{\text{fix}}(\phi) + T(\mathbb{P}_{\leq n-1},\mathbb{V}^{\text{fix}})[\phi])})}$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathcal{V}' \to \mathcal{V}'$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathcal{V}'$~$ |
---|

Existence Proof of Logical Inductor | $~$[0,1]^{S'}$~$ |
---|

Existence Proof of Logical Inductor | $~$x$~$ |
---|

Existence Proof of Logical Inductor | $~$f(x)=x$~$ |
---|

Existence Proof of Logical Inductor | $~$\text{fix}\mathbb{V}(\phi)$~$ |
---|

Existence Proof of Logical Inductor | $~$T$~$ |
---|

Existence Proof of Logical Inductor | $~$\mathbb{P}$~$ |
---|

Existence Proof of Logical Inductor | $~$T$~$ |
---|

Existence Proof of Logical Inductor | $~$1-2^{-n}$~$ |
---|

Existence Proof of Logical Inductor | $~$2^{-n}$~$ |
---|

Existence Proof of Logical Inductor | $~$B(n,b, T_n, \mathbb{P}_{\leq n-1})$~$ |
---|

Existence Proof of Logical Inductor | $~$(n-1)$~$ |
---|

Existence Proof of Logical Inductor | $~$m<n$~$ |
---|

Expected value | $~$V = x_{1},$~$ |
---|

Expected value | $~$V = x_{2}, …, $~$ |
---|

Expected value | $~$V = x_{k}$~$ |
---|

Expected value | $~$P(x_{i})$~$ |
---|

Expected value | $~$V = x_{i}$~$ |
---|

Expected value | $$~$\sum_{i=1}^{k}x_{i}P(x_{i})$~$$ |
---|

Expected value | $~$x \in \mathbb{R}$~$ |
---|

Expected value | $~$P(x)$~$ |
---|

Expected value | $~$\lim_{dx \to 0}$~$ |
---|

Expected value | $~$x<V<(x+dx)$~$ |
---|

Expected value | $~$dx$~$ |
---|

Expected value | $$~$\int_{-∞}^{∞}xP(x)dx$~$$ |
---|

Explicit Bayes as a counter for 'worrying' | $~$\mathbb P(\text{cancel}|\text{desirable})$~$ |
---|

Explicit Bayes as a counter for 'worrying' | $~$\mathbb P(\text{cancel}|\text{undesirable})$~$ |
---|

Exponential | $~$b$~$ |
---|

Exponential | $~$x$~$ |
---|

Exponential | $~$b^x,$~$ |
---|

Exponential | $~$b$~$ |
---|

Exponential | $~$x$~$ |
---|

Exponential | $~$10^3$~$ |
---|

Exponential | $~$10 \cdot 10 \cdot 10 = 1000$~$ |
---|

Exponential | $~$2^4=16,$~$ |
---|

Exponential | $~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$ |
---|

Exponential | $~$x$~$ |
---|

Exponential | $~$10^{1/2}$~$ |
---|

Exponential | $~$n$~$ |
---|

Exponential | $~$n$~$ |
---|

Exponential | $~$n \approx 3.16,$~$ |
---|

Exponential | $~$n \cdot n \approx 10.$~$ |
---|

Exponential | $~$f(x) = c \times a^x$~$ |
---|

Exponential | $~$c$~$ |
---|

Exponential | $~$a$~$ |
---|

Exponential | $~$1.02$~$ |
---|

Exponential | $~$f(x) = 100 \times 1.02^x$~$ |
---|

Exponential | $~$x$~$ |
---|

Exponential | $~$x$~$ |
---|

Exponential | $~$f(x) = 1 \times 2^x$~$ |
---|

Exponential | $~$f(x) = f(x-1) \times 1.02$~$ |
---|

Exponential | $~$\Delta f(x) = f(x+1) - f(x) = 0.02 \times f(x)$~$ |
---|

Exponential | $~$f(x) = f(x-1) + 0.02 \times f(0)$~$ |
---|

Exponential | $~$f(0)$~$ |
---|

Exponential | $~$f(x)$~$ |
---|

Exponential notation for function spaces | $~$X$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$X$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$X \to Y$~$ |
---|

Exponential notation for function spaces | $~$Y^X$~$ |
---|

Exponential notation for function spaces | $~$Y^3$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$f : X \to Y$~$ |
---|

Exponential notation for function spaces | $~$X$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$X$~$ |
---|

Exponential notation for function spaces | $~$Y^n$~$ |
---|

Exponential notation for function spaces | $~$n$~$ |
---|

Exponential notation for function spaces | $~$Y$~$ |
---|

Exponential notation for function spaces | $~$|X| = n$~$ |
---|

Exponential notation for function spaces | $~$Y^X \cong Y^n$~$ |
---|

Exponential notation for function spaces | $~$Z^{X \times Y} \cong (Z^X)^Y$~$ |
---|

Exponential notation for function spaces | $~$Z^{X + Y} \cong Z^X \times Z^Y$~$ |
---|

Exponential notation for function spaces | $~$Z^1 \cong Z$~$ |
---|

Exponential notation for function spaces | $~$1$~$ |
---|

Exponential notation for function spaces | $~$Z$~$ |
---|

Exponential notation for function spaces | $~$Z$~$ |
---|

Exponential notation for function spaces | $~$Z^0 \cong 1$~$ |
---|

Exponential notation for function spaces | $~$0$~$ |
---|

Exponential notation for function spaces | $~$Y^X$~$ |
---|

Exponential notation for function spaces | $~$\text{Hom}_{\mathcal{C}}(X, Y)$~$ |
---|

Exponential notation for function spaces | $~$\mathcal{C}$~$ |
---|

Extensionality Axiom | $$~$ \forall A \forall B : ( \forall x : (x \in A \iff x \in B) \Rightarrow A=B)$~$$ |
---|

Extensionality Axiom | $~$\{1,2\} = \{2,1\}$~$ |
---|

Extensionality Axiom | $~$1$~$ |
---|

Extensionality Axiom | $~$2$~$ |
---|

Extensionality Axiom | $~$5$~$ |
---|

Extensionality Axiom | $~$73$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$(1 : 9 ) \times (3 : 1) \ = \ (3 : 9) \ \cong \ (1 : 3)$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$X$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$X$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$X.$~$ |
---|

Extraordinary claims require extraordinary evidence | $$~$\text{Likelihood ratio} = \dfrac{\text{Probability of seeing the evidence, assuming the claim is true}}{\text{Probability of seeing the evidence, assuming the claim is false}}$~$$ |
---|

Extraordinary claims require extraordinary evidence | $~$10^{100}$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$10^{94}$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$(10^{94} : 1)$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$10^{-94}$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$(1 : 10^{100})$~$ |
---|

Extraordinary claims require extraordinary evidence | $~$(1 : 10^6)$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$1,2,3$~$ |
---|

Factorial | $~$1,2,3$~$ |
---|

Factorial | $~$1,3,2$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$2$~$ |
---|

Factorial | $~$3$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$1,2,3$~$ |
---|

Factorial | $~$1,3,2$~$ |
---|

Factorial | $~$2,1,3$~$ |
---|

Factorial | $~$2,3,1$~$ |
---|

Factorial | $~$3,1,2$~$ |
---|

Factorial | $~$3,2,1$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$2$~$ |
---|

Factorial | $~$3$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$1,2,3,4$~$ |
---|

Factorial | $~$1,2,4,3$~$ |
---|

Factorial | $~$1,3,2,4$~$ |
---|

Factorial | $~$1,3,4,2$~$ |
---|

Factorial | $~$1,4,2,3$~$ |
---|

Factorial | $~$1,4,3,2$~$ |
---|

Factorial | $~$2,1,3,4$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$2$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$3$~$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$4$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$120$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$2$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$3$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$4$~$ |
---|

Factorial | $~$24$~$ |
---|

Factorial | $~$5$~$ |
---|

Factorial | $~$120$~$ |
---|

Factorial | $~$5$~$ |
---|

Factorial | $~$4$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n-1$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n-1$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$2$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n-1$~$ |
---|

Factorial | $~$n-1$~$ |
---|

Factorial | $~$5!$~$ |
---|

Factorial | $~$120$~$ |
---|

Factorial | $~$4!$~$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$5! = 5 \times 4!$~$ |
---|

Factorial | $~$4! = 4 \times 3!$~$ |
---|

Factorial | $~$5$~$ |
---|

Factorial | $~$n-1$~$ |
---|

Factorial | $~$n \times n - 1!$~$ |
---|

Factorial | $~$(n \times n)-1!$~$ |
---|

Factorial | $$~$n! = n \times (n-1)!$~$$ |
---|

Factorial | $~$n! = n \times (n-1)!$~$ |
---|

Factorial | $~$(n-1)! = (n-1) \times (n-2)!$~$ |
---|

Factorial | $~$(n-2)! = (n-2) \times (n-3)!$~$ |
---|

Factorial | $$~$n! = n \times (n-1)! = n \times (n-1) \times (n-2)! = n \times (n-1) \times (n-2) \times (n-3)!$~$$ |
---|

Factorial | $$~$n \times (n-1) \times (n-2) \times \dots \times 5 \times 4 \times 3!$~$$ |
---|

Factorial | $~$3! = 6$~$ |
---|

Factorial | $~$3 \times 2 \times 1$~$ |
---|

Factorial | $$~$n! = n \times (n-1) \times \dots \times 4 \times 3 \times 2 \times 1$~$$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$3!$~$ |
---|

Factorial | $~$2!$~$ |
---|

Factorial | $~$1!$~$ |
---|

Factorial | $~$1,2$~$ |
---|

Factorial | $~$2,1$~$ |
---|

Factorial | $~$2! = 2$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$1! = 1$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$0! = 1$~$ |
---|

Factorial | $~$5!$~$ |
---|

Factorial | $~$1*2*3*4*5$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n!=\prod_{i=1}^{n}i$~$ |
---|

Factorial | $~$0! = 1$~$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$A$~$ |
---|

Factorial | $~$B$~$ |
---|

Factorial | $~$C$~$ |
---|

Factorial | $$~$ABC$~$$ |
---|

Factorial | $$~$ACB$~$$ |
---|

Factorial | $$~$BAC$~$$ |
---|

Factorial | $$~$BCA$~$$ |
---|

Factorial | $$~$CAB$~$$ |
---|

Factorial | $$~$CBA$~$$ |
---|

Factorial | $~$6$~$ |
---|

Factorial | $~$3$~$ |
---|

Factorial | $~$6 = 3*2*1 = 3!$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n+1$~$ |
---|

Factorial | $~$1$~$ |
---|

Factorial | $$~$A$~$$ |
---|

Factorial | $$~$1 = \prod_{i=1}^{1}i = 1!$~$$ |
---|

Factorial | $~$\{A_{1},A_{2},…,A_{n},A_{n+1}\}$~$ |
---|

Factorial | $~$n+1$~$ |
---|

Factorial | $~$A_{n+1}$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$A_{n+1}$~$ |
---|

Factorial | $~$A_{n+1}$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n$~$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$A_{n+1}$~$ |
---|

Factorial | $~$n!$~$ |
---|

Factorial | $~$A_{n+1}$~$ |
---|

Factorial | $~$n!*(n+1)$~$ |
---|

Factorial | $~$(n+1)!$~$ |
---|

Factorial | $~$x!$~$ |
---|

Factorial | $$~$x! = \Gamma (x+1),$~$$ |
---|

Factorial | $~$\Gamma $~$ |
---|

Factorial | $$~$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $~$x$~$ |
---|

Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $~$x=1$~$ |
---|

Factorial | $$~$\prod_{i=1}^{1}i = \int_{0}^{\infty}t^{1}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$1=1$~$$ |
---|

Factorial | $~$x$~$ |
---|

Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $~$x + 1$~$ |
---|

Factorial | $$~$\prod_{i=1}^{x+1}i = \int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $~$x+1$~$ |
---|

Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$(x+1)\prod_{i=1}^{x}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$\prod_{i=1}^{x+1}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$= 0+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|

Factorial | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}-\int_{0}^{\infty}(x+1)t^{x}(-e^{-t})\mathrm{d} t$~$$ |
---|

Factorial | $$~$=\int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|

Faithful simulation | $~$D$~$ |
---|

Faithful simulation | $~$S_D$~$ |
---|

Faithful simulation | $~$D$~$ |
---|

Faithful simulation | $~$D$~$ |
---|

Faithful simulation | $~$S_D$~$ |
---|

Faithful simulation | $~$D.$~$ |
---|

Field homomorphism is trivial or injective | $~$F$~$ |
---|

Field homomorphism is trivial or injective | $~$G$~$ |
---|

Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$0$~$ |
---|

Field homomorphism is trivial or injective | $~$0$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$0$~$ |
---|

Field homomorphism is trivial or injective | $~$F$~$ |
---|

Field homomorphism is trivial or injective | $~$G$~$ |
---|

Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$0$~$ |
---|

Field homomorphism is trivial or injective | $~$x \in F$~$ |
---|

Field homomorphism is trivial or injective | $~$f(x) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$x,y$~$ |
---|

Field homomorphism is trivial or injective | $~$f(x) = f(y)$~$ |
---|

Field homomorphism is trivial or injective | $~$x = y$~$ |
---|

Field homomorphism is trivial or injective | $~$f(x) = f(y)$~$ |
---|

Field homomorphism is trivial or injective | $~$f(x)-f(y) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f(x-y) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$z = 0_F$~$ |
---|

Field homomorphism is trivial or injective | $~$z = x-y$~$ |
---|

Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$z$~$ |
---|

Field homomorphism is trivial or injective | $~$0_F$~$ |
---|

Field homomorphism is trivial or injective | $~$z^{-1}$~$ |
---|

Field homomorphism is trivial or injective | $~$f(z^{-1}) f(z) = f(z^{-1}) \times 0_G = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$f(z^{-1} \times z) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f(1_F) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$f$~$ |
---|

Field homomorphism is trivial or injective | $~$F \setminus \{ 0_F \}$~$ |
---|

Field homomorphism is trivial or injective | $~$G \setminus \{0_G\}$~$ |
---|

Field homomorphism is trivial or injective | $~$1_F$~$ |
---|

Field homomorphism is trivial or injective | $~$F \setminus \{0_F\}$~$ |
---|

Field homomorphism is trivial or injective | $~$1_G$~$ |
---|

Field homomorphism is trivial or injective | $~$F \setminus \{0_G\}$~$ |
---|

Field homomorphism is trivial or injective | $~$z$~$ |
---|

Field homomorphism is trivial or injective | $~$z \not = 0_F$~$ |
---|

Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|

Field homomorphism is trivial or injective | $~$z = 0_F$~$ |
---|

Field structure of rational numbers | $~$\frac{a}{b} + \frac{p}{q} = \frac{aq+bp}{bq}$~$ |
---|

Field structure of rational numbers | $~$\frac{a}{b} \frac{c}{d} = \frac{ac}{bd}$~$ |
---|

Field structure of rational numbers | $~$\frac{0}{1}$~$ |
---|

Field structure of rational numbers | $~$\frac{1}{1}$~$ |
---|

Field structure of rational numbers | $~$\frac{a}{b}$~$ |
---|

Field structure of rational numbers | $~$\frac{-a}{b}$~$ |
---|

Field structure of rational numbers | $~$\frac{a}{b}$~$ |
---|

Field structure of rational numbers | $~$a \not = 0$~$ |
---|

Field structure of rational numbers | $~$\frac{b}{a}$~$ |
---|

Field structure of rational numbers | $~$0 < \frac{c}{d}$~$ |
---|

Field structure of rational numbers | $~$c$~$ |
---|

Field structure of rational numbers | $~$d$~$ |
---|

Field structure of rational numbers | $~$c$~$ |
---|

Field structure of rational numbers | $~$d$~$ |
---|

Field structure of rational numbers | $~$\frac{a}{b} < \frac{c}{d}$~$ |
---|

Field structure of rational numbers | $~$0 < \frac{c}{d} - \frac{a}{b}$~$ |
---|

Finite set | $~$X$~$ |
---|

Finite set | $~$n \in \mathbb{N}$~$ |
---|

Finite set | $~$X$~$ |
---|

Finite set | $~$n$~$ |
---|

Finite set | $~$\{ 1,2 \}$~$ |
---|

Finite set | $~$\{ \mathbb{N} \}$~$ |
---|

Finite set | $~$\mathbb{N}$~$ |
---|

Finite set | $~$\mathbb{R}$~$ |
---|

First order linear equations | $$~$
u'=a(t)u+b(t)
$~$$ |
---|

First order linear equations | $~$a$~$ |
---|

First order linear equations | $~$b$~$ |
---|

First order linear equations | $~$[\alpha, \beta]$~$ |
---|

First order linear equations | $~$b$~$ |
---|

First order linear equations | $~$b=0$~$ |
---|

First order linear equations | $$~$
u'=a(t)u
$~$$ |
---|

First order linear equations | $~$C^1$~$ |
---|

First order linear equations | $~$[\alpha, \beta]$~$ |
---|

First order linear equations | $~$b$~$ |
---|

First order linear equations | $~$\Sigma_b$~$ |
---|

First order linear equations | $~$\Sigma_0$~$ |
---|

First order linear equations | $~$\Sigma_0$~$ |
---|

First order linear equations | $~$\Sigma_b$~$ |
---|

First order linear equations | $~$\Sigma_0$~$ |
---|

First order linear equations | $~$\Sigma_0$~$ |
---|

First order linear equations | $~$\Sigma_0$~$ |
---|

First order linear equations | $~$\Sigma_b$~$ |
---|

First order linear equations | $~$\Sigma_b$~$ |
---|

First order linear equations | $~$a$~$ |
---|

First order linear equations | $~$b$~$ |
---|

First order linear equations | $$~$
u' = au+b
$~$$ |
---|

First order linear equations | $~$u'=au$~$ |
---|

First order linear equations | $~$ke^{\int_{t_0}^ta}$~$ |
---|

First order linear equations | $~$k$~$ |
---|

First order linear equations | $~$t_0\in [\alpha, \beta]$~$ |
---|

First order linear equations | $~$u=h\dot v$~$ |
---|

First order linear equations | $~$h$~$ |
---|

First order linear equations | $~$e^{\int_{t_0}^ta}$~$ |
---|

First order linear equations | $~$u$~$ |
---|

First order linear equations | $$~$
u'=(hv)'=h'v+hv'=au+b=a(hv)+b
$~$$ |
---|

First order linear equations | $~$h\in\Sigma_0$~$ |
---|

First order linear equations | $~$h'=ah$~$ |
---|

First order linear equations | $$~$
v'=bh^{-1}=be^{-\int_{t_0}^ta}
$~$$ |
---|

First order linear equations | $$~$
v=\int_{t_0}^tbe^{\int_{t}^sa}ds
$~$$ |
---|

First order linear equations | $~$\Sigma_b$~$ |
---|

First order linear equations | $~$ke^{\int_{t_0}^ta}+\int_{t_0}^tbe^{\int_{t}^sa}ds$~$ |
---|

First order linear equations | $~$k$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p, q_1,…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$H(q_1,..,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p,q_1,…,q_n)] \leftrightarrow \boxdot[p\leftrightarrow H(q_1,..,q_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p)] \leftrightarrow \boxdot[p\leftrightarrow H]$~$ |
---|

Fixed point theorem of provability logic | $~$\boxdot A = A\wedge \square A$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\psi(p, q_1…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$H(q_1,…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow\psi(p, q_1,…,q_n)] \leftrightarrow \boxdot[p_i\leftrightarrow H(q_1,…,q_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$\psi$~$ |
---|

Fixed point theorem of provability logic | $~$\psi$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash H(q_1,…,q_n)\leftrightarrow \phi(H(q_1,…,q_n),q_1,…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow\psi(p, q_1,…,q_n)] \leftrightarrow \boxdot[p_i\leftrightarrow H(q_1,…,q_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$GL$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow\psi(H(q_1,…,q_n), q_1,…,q_n)] \leftrightarrow \boxdot[H(q_1,…,q_n)\leftrightarrow H(q_1,…,q_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow H(q_1,…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow\psi(H(q_1,…,q_n), q_1,…,q_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$I$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash H\leftrightarrow I$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow \phi(p))\leftrightarrow (p\leftrightarrow H)$~$ |
---|

Fixed point theorem of provability logic | $~$I$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash H\leftrightarrow I$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash F(I)\leftrightarrow F(H)$~$ |
---|

Fixed point theorem of provability logic | $~$F(q)$~$ |
---|

Fixed point theorem of provability logic | $~$F(q)=\boxdot(p\leftrightarrow q)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow H)\leftrightarrow \boxdot(p\leftrightarrow I)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow \phi(p))\leftrightarrow (p\leftrightarrow I)$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$I$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot (p\leftrightarrow H)\leftrightarrow \boxdot (p\leftrightarrow I)$~$ |
---|

Fixed point theorem of provability logic | $~$GL$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash\boxdot (H\leftrightarrow H)\leftrightarrow \boxdot (H\leftrightarrow I)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot (H\leftrightarrow H)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash (H\leftrightarrow I)$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p)] \leftrightarrow \boxdot[p\leftrightarrow H]$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$\square^n \bot$~$ |
---|

Fixed point theorem of provability logic | $~$\square^n A = \underbrace{\square,\square,\ldots,\square}_{n\text{-times}} A$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$B$~$ |
---|

Fixed point theorem of provability logic | $~$[[B]]_A$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$[[\bot]]_A = \emptyset$~$ |
---|

Fixed point theorem of provability logic | $~$[[B\to C]]_A = (\mathbb{N} \setminus [[B]]_A)\cup [[C]]_A$~$ |
---|

Fixed point theorem of provability logic | $~$[[\square D]]_A=\{m:\forall i < m i\in [[D]]_A\}$~$ |
---|

Fixed point theorem of provability logic | $~$[[p]]_A=[[A]]_A$~$ |
---|

Fixed point theorem of provability logic | $~$M$~$ |
---|

Fixed point theorem of provability logic | $~$(p\leftrightarrow A) is valid, and $~$ |
---|

Fixed point theorem of provability logic | $~$ a $~$ |
---|

Fixed point theorem of provability logic | $~$-sentence. Then $~$ |
---|

Fixed point theorem of provability logic | $~$ iff $~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$B$~$ |
---|

Fixed point theorem of provability logic | $~$n$~$ |
---|

Fixed point theorem of provability logic | $~$n$~$ |
---|

Fixed point theorem of provability logic | $~$n$~$ |
---|

Fixed point theorem of provability logic | $~$\square$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$A$~$ |
---|

Fixed point theorem of provability logic | $~$p\leftrightarrow A$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$\square^{n+1}\bot\wedge \square^n \bot$~$ |
---|

Fixed point theorem of provability logic | $~$n$~$ |
---|

Fixed point theorem of provability logic | $~$p\leftrightarrow \neg\square p$~$ |
---|

Fixed point theorem of provability logic | $~$\neg\square p$~$ |
---|

Fixed point theorem of provability logic | $~$0$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\square B$~$ |
---|

Fixed point theorem of provability logic | $~$0$~$ |
---|

Fixed point theorem of provability logic | $~$B$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\neg\square p$~$ |
---|

Fixed point theorem of provability logic | $$~$
\begin{array}{cccc}
\text{world= } & p & \square (p) & \neg \square (p) \\
0 & \bot & \top & \bot \\
1 & \top & \bot & \top \\
2 & \top & \bot & \top \\
\end{array}
$~$$ |
---|

Fixed point theorem of provability logic | $~$\square$~$ |
---|

Fixed point theorem of provability logic | $~$2$~$ |
---|

Fixed point theorem of provability logic | $~$[[p]]_{\neg\square p} = \mathbb{N}\setminus \{0\}$~$ |
---|

Fixed point theorem of provability logic | $~$H = \square^{0+1}\bot \wedge \square^0\bot = \neg\square\bot$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \square [p\leftrightarrow \neg\square p]\leftrightarrow \square[p\leftrightarrow \neg\square \bot]$~$ |
---|

Fixed point theorem of provability logic | $~$PA$~$ |
---|

Fixed point theorem of provability logic | $~$PA\vdash \square_{PA} [G\leftrightarrow \neg\square_{PA} G]\leftrightarrow \square_{PA}[G\leftrightarrow \neg\square_{PA} \bot]$~$ |
---|

Fixed point theorem of provability logic | $~$G$~$ |
---|

Fixed point theorem of provability logic | $~$PA$~$ |
---|

Fixed point theorem of provability logic | $~$G$~$ |
---|

Fixed point theorem of provability logic | $~$PA\vdash G\leftrightarrow \neg\square_{PA} G$~$ |
---|

Fixed point theorem of provability logic | $~$G$~$ |
---|

Fixed point theorem of provability logic | $~$PA\vdash \square_PA[ G\leftrightarrow \neg\square_{PA} G]$~$ |
---|

Fixed point theorem of provability logic | $~$PA\vdash \square_{PA}[G\leftrightarrow \neg\square_{PA} \bot]$~$ |
---|

Fixed point theorem of provability logic | $~$PA$~$ |
---|

Fixed point theorem of provability logic | $~$PA\vdash G\leftrightarrow \neg\square_{PA} \bot$~$ |
---|

Fixed point theorem of provability logic | $~$G$~$ |
---|

Fixed point theorem of provability logic | $~$PA$~$ |
---|

Fixed point theorem of provability logic | $~$\omega$~$ |
---|

Fixed point theorem of provability logic | $~$H\leftrightarrow\square H$~$ |
---|

Fixed point theorem of provability logic | $$~$
\begin{array}{ccc}
\text{world= } & p & \square (p) \\
0 & \top & \top \\
1 & \top & \top \\
\end{array}
$~$$ |
---|

Fixed point theorem of provability logic | $~$\top$~$ |
---|

Fixed point theorem of provability logic | $~$\phi(p, q_1,…,q_n)$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$B(\square D_1(p), …, \square D_{k}(p))$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$\square$~$ |
---|

Fixed point theorem of provability logic | $~$q_i$~$ |
---|

Fixed point theorem of provability logic | $~$B$~$ |
---|

Fixed point theorem of provability logic | $~$D_i$~$ |
---|

Fixed point theorem of provability logic | $~$k$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$0$~$ |
---|

Fixed point theorem of provability logic | $~$p$~$ |
---|

Fixed point theorem of provability logic | $~$B_i = B(\square D_1(p), …, \square D_{i-1}(p),\top, \square D_{i+1}(p),…,\square D_k(p))$~$ |
---|

Fixed point theorem of provability logic | $~$k-1$~$ |
---|

Fixed point theorem of provability logic | $~$k-1$~$ |
---|

Fixed point theorem of provability logic | $~$H_i$~$ |
---|

Fixed point theorem of provability logic | $~$B_i$~$ |
---|

Fixed point theorem of provability logic | $~$H=B(\square D_1(H_1),…,\square D_k(H_k))$~$ |
---|

Fixed point theorem of provability logic | $~$\phi$~$ |
---|

Fixed point theorem of provability logic | $~$p\leftrightarrow \neg\square(q\to p)$~$ |
---|

Fixed point theorem of provability logic | $~$B(d)=\neg d$~$ |
---|

Fixed point theorem of provability logic | $~$D_1(p)=q\to p$~$ |
---|

Fixed point theorem of provability logic | $~$B_1(p)=\neg \top = \bot$~$ |
---|

Fixed point theorem of provability logic | $~$H=B(\square D_1(\bot))=\neg\square \neg q$~$ |
---|

Fixed point theorem of provability logic | $~$p\leftrightarrow \square [\square(p\wedge q)\wedge \square(p\wedge r)]$~$ |
---|

Fixed point theorem of provability logic | $~$B(a)=a$~$ |
---|

Fixed point theorem of provability logic | $~$D_1(p)=\square(p\wedge q)\wedge \square(p\wedge r)$~$ |
---|

Fixed point theorem of provability logic | $~$B(\top)$~$ |
---|

Fixed point theorem of provability logic | $~$\top$~$ |
---|

Fixed point theorem of provability logic | $~$B(\square D_1(p=\top))=\square[\square(\top\wedge q)\wedge \square(\top\wedge r)]=\square[\square(q)\wedge \square(r)]$~$ |
---|

Fixed point theorem of provability logic | $~$A_i(p_1,…,p_n)$~$ |
---|

Fixed point theorem of provability logic | $~$n$~$ |
---|

Fixed point theorem of provability logic | $~$A_i$~$ |
---|

Fixed point theorem of provability logic | $~$p_n$~$ |
---|

Fixed point theorem of provability logic | $~$p_js$~$ |
---|

Fixed point theorem of provability logic | $~$H_1, …,H_n$~$ |
---|

Fixed point theorem of provability logic | $~$p_j$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le n} \{\boxdot (p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le n} \{\boxdot(p_i\leftrightarrow H_i)\}$~$ |
---|

Fixed point theorem of provability logic | $~$H$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_1\leftrightarrow A_i(p_1,…,p_n)) \leftrightarrow \boxdot(p_1\leftrightarrow H(p_2,…,p_n))$~$ |
---|

Fixed point theorem of provability logic | $~$j$~$ |
---|

Fixed point theorem of provability logic | $~$H_1,…,H_j$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n))\}$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(A\leftrightarrow B)\rightarrow [F(A)\leftrightarrow F(B)]$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n)\rightarrow [\boxdot(p_{j+1}\leftrightarrow A_{j+1}(p_{1},…,p_n))\leftrightarrow \boxdot(p_{j+1}\leftrightarrow A_{j+1}(p_{1},…,p_{i-1},H_i(p_{j+1},…,p_n),p_{i+1},…,p_n))]$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\rightarrow \boxdot(p_{j+1}\leftrightarrow A_{j+1}(H_1,…,H_j,p_{j+1},…,p_n))$~$ |
---|

Fixed point theorem of provability logic | $~$H_{j+1}'$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_{j+1}\leftrightarrow A_{j+1}(H_1,…,H_j,p_{j+1},…,p_n)) \leftrightarrow \boxdot[p_{j+1}\leftrightarrow H_{j+1}'(p_{j+2},…,p_n)]$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p_{j+1}\leftrightarrow H_{j+1}'(p_{j+2},…,p_n)]\rightarrow [\boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n)) \leftrightarrow \boxdot(p_i\leftrightarrow H_i(H_{j+1}',…,p_n))$~$ |
---|

Fixed point theorem of provability logic | $~$H_{i}'$~$ |
---|

Fixed point theorem of provability logic | $~$H_i(H_{j+1}',…,p_n)$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\rightarrow \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow H_i'(p_{j+2},…,p_n))\}$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow H_i'(p_{j+2},…,p_n))\}$~$ |
---|

Fixed point theorem of provability logic | $~$\square$~$ |
---|

Fixed point theorem of provability logic | $~$H_i'$~$ |
---|

Fixed point theorem of provability logic | $~$H_i$~$ |
---|

Fixed point theorem of provability logic | $~$A_i$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash H_i\leftrightarrow A_i(H_1,…,H_n)$~$ |
---|

Fixed point theorem of provability logic | $~$GL$~$ |
---|

Fixed point theorem of provability logic | $~$p_i$~$ |
---|

Fixed point theorem of provability logic | $~$H_i$~$ |
---|

Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le n} \{\boxdot (H_i\leftrightarrow A_i(H_1,…,H_n)\}\leftrightarrow \wedge_{i\le n} \{\boxdot(H_i\leftrightarrow H_i)\}$~$ |
---|

Fixed point theorem of provability logic | $~$GL$~$ |
---|

Flag the load-bearing premises | $~$\neg X$~$ |
---|

Formal Logic | $~$S$~$ |
---|

Formal Logic | $~$O$~$ |
---|

Formal Logic | $~$M$~$ |
---|

Formal Logic | $~$C$~$ |
---|

Formal Logic | $~$S$~$ |
---|

Formal Logic | $~$O$~$ |
---|

Formal Logic | $~$S$~$ |
---|

Formal Logic | $~$O$~$ |
---|

Formal Logic | $~$M$~$ |
---|

Formal Logic | $~$C$~$ |
---|

Formal Logic | $~$M$~$ |
---|

Formal Logic | $~$C$~$ |
---|

Formal Logic | $~$\rightarrow$~$ |
---|

Formal Logic | $~$A$~$ |
---|

Formal Logic | $~$B$~$ |
---|

Formal Logic | $~$A \rightarrow B$~$ |
---|

Formal Logic | $~$\therefore$~$ |
---|

Formal definition of the free group | $~$X^r$~$ |
---|

Formal definition of the free group | $~$X \cup X^{-1}$~$ |
---|

Formal definition of the free group | $~$aa^{-1}$~$ |
---|

Formal definition of the free group | $~$r$~$ |
---|

Formal definition of the free group | $~$F(X)$~$ |
---|

Formal definition of the free group | $~$FX$~$ |
---|

Formal definition of the free group | $~$X$~$ |
---|

Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|

Formal definition of the free group | $~$x \in X \cup X^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_x : \mathrm{Sym}(X^r) \to \mathrm{Sym}(X^r)$~$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_n \mapsto a_1 a_2 \dots a_n x$~$ |
---|

Formal definition of the free group | $~$a_n \not = x^{-1}$~$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_{n-1} x^{-1} \mapsto a_1 a_2 \dots a_{n-1}$~$ |
---|

Formal definition of the free group | $~$\rho_{x^{-1}} : \mathrm{Sym}(X^r) \to \mathrm{Sym}(X^r)$~$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_n \mapsto a_1 a_2 \dots a_n x^{-1}$~$ |
---|

Formal definition of the free group | $~$a_n \not = x$~$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_{n-1} x \mapsto a_1 a_2 \dots a_{n-1}$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|

Formal definition of the free group | $~$X^r$~$ |
---|

Formal definition of the free group | $~$X^r$~$ |
---|

Formal definition of the free group | $~$X^r$~$ |
---|

Formal definition of the free group | $~$X$~$ |
---|

Formal definition of the free group | $~$x^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$x$~$ |
---|

Formal definition of the free group | $~$x^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$x^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$x$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$X^r \to X^r$~$ |
---|

Formal definition of the free group | $~$x^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|

Formal definition of the free group | $~$\rho_{\varepsilon}$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|

Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|

Formal definition of the free group | $~$\rho_x \cdot \rho_y = \rho_x \circ \rho_y$~$ |
---|

Formal definition of the free group | $~$\rho_x \rho_y$~$ |
---|

Formal definition of the free group | $~$\rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_1}$~$ |
---|

Formal definition of the free group | $~$\varepsilon$~$ |
---|

Formal definition of the free group | $$~$\rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_1}(\varepsilon) = \rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_3}(\rho_{a_2}(a_1)) = \rho_{a_n a_{n-1} \dots a_3}(a_1 a_2) = \dots = a_1 a_2 \dots a_n$~$$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_n$~$ |
---|

Formal definition of the free group | $~$\rho_{a_i}, \rho_{a_{i+1}}$~$ |
---|

Formal definition of the free group | $~$\rho_{a_i}$~$ |
---|

Formal definition of the free group | $~$w = a_1 a_2 \dots a_n$~$ |
---|

Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}$~$ |
---|

Formal definition of the free group | $~$\rho_{a_1} \circ \rho_{a_2} \circ \dots \circ \rho_{a_n}$~$ |
---|

Formal definition of the free group | $~$a_i$~$ |
---|

Formal definition of the free group | $~$X \cup X^{-1}$~$ |
---|

Formal definition of the free group | $~$\rho_{a_i}$~$ |
---|

Formal definition of the free group | $~$a_1 a_2 \dots a_n$~$ |
---|

Formal definition of the free group | $~$b_1 b_2 \dots b_m$~$ |
---|

Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n} = \rho_{b_1} \rho_{b_2} \dots \rho_{b_m}$~$ |
---|

Formal definition of the free group | $~$a_1 \dots a_n = b_1 \dots b_m$~$ |
---|

Formal definition of the free group | $~$\varepsilon$~$ |
---|

Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}$~$ |
---|

Formal definition of the free group | $~$a_n a_{n-1} \dots a_2 a_1$~$ |
---|

Formal definition of the free group | $~$\rho_{b_1} \rho_{b_2} \dots \rho_{b_m}$~$ |
---|

Formal definition of the free group | $~$b_m b_{m-1} \dots b_2 b_1$~$ |
---|

Formal definition of the free group | $~$\rho_x$~$ |
---|

Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|

Formal definition of the free group | $~$x \in X$~$ |
---|

Formal definition of the free group | $~$\rho_{x_1} \dots \rho_{x_n}$~$ |
---|

Formal definition of the free group | $~$x_1, \dots, x_n \in X \cup X^{-1}$~$ |
---|

Formal definition of the free group | $~$x_1 \dots x_n$~$ |
---|

Formal definition of the free group | $~$x_i, x_{i+1}$~$ |
---|

Formal definition of the free group | $~$\rho_{x_1} \dots \rho_{x_n}$~$ |
---|

Formal definition of the free group | $~$\rho_{x_1} \rho_{x_1^{-1}} \rho_{x_2} = \rho_{x_2}$~$ |
---|

Fractional bits | $~$\log_2(8) = 3$~$ |
---|

Fractional bits | $~$\log_2(1024) = 10$~$ |
---|

Fractional bits | $~$\log_2(3) \approx 1.58.$~$ |
---|

Fractional bits | $~$\log_2(3),$~$ |
---|

Fractional bits | $~$n \ge 5$~$ |
---|

Fractional bits | $~$n - 5.$~$ |
---|

Fractional bits: Digit usage interpretation | $~$10 \cdot 10 \cdot \sqrt{10} \approx 316,$~$ |
---|

Fractional bits: Digit usage interpretation | $~$\sqrt{10}$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(7)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$n$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\lceil \log_2(n) \rceil$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(7) \neq 2.875,$~$ |
---|

Fractional bits: Expected cost interpretation | $~$(m, n)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$7m + n,$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\lceil \log_2(49) \rceil = 6$~$ |
---|

Fractional bits: Expected cost interpretation | $~$64 - 49 = 15$~$ |
---|

Fractional bits: Expected cost interpretation | $~$6 - \frac{15}{49} \approx 5.694$~$ |
---|

Fractional bits: Expected cost interpretation | $~$(9 - \frac{169}{343})\approx 8.507$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\approx 2.836$~$ |
---|

Fractional bits: Expected cost interpretation | $~$2.807$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(7)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$n$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\lceil \log_2(n) \rceil$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(n).$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(n)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(n)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b$~$ |
---|

Fractional bits: Expected cost interpretation | $~$x < \log_2(b)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_b(2) \cdot x$~$ |
---|

Fractional bits: Expected cost interpretation | $~$2$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_b(2)$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b$~$ |
---|

Fractional bits: Expected cost interpretation | $~$2$~$ |
---|

Fractional bits: Expected cost interpretation | $~$x$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_b(2) \cdot \log_2(b) = 1$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b,$~$ |
---|

Fractional bits: Expected cost interpretation | $~$b$~$ |
---|

Fractional bits: Expected cost interpretation | $~$\log_2(b)$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$\log_b(x)$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$\log_{3.16}(5.62) \approx 1.5$~$ |
---|

Fractional digits | $~$3.16^{1.5} \approx 5.62,$~$ |
---|

Fractional digits | $~$a$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$5a + b.$~$ |
---|

Fractional digits | $~$\log_{10}(5) + \log_{10}(2) = 1$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$y$~$ |
---|

Fractional digits | $~$x \cdot y \le n$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$y$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$18$~$ |
---|

Fractional digits | $~$3$~$ |
---|

Fractional digits | $~$6$~$ |
---|

Fractional digits | $~$a$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$6a+b.$~$ |
---|

Fractional digits | $~$n = x \cdot y,$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$y$~$ |
---|

Fractional digits | $~$n = x \cdot y$~$ |
---|

Fractional digits | $~$\log_b(x) + \log_b(y) = \log_b(n),$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x \cdot x < 10.$~$ |
---|

Fractional digits | $~$a$~$ |
---|

Fractional digits | $~$b$~$ |
---|

Fractional digits | $~$31a + b$~$ |
---|

Fractional digits | $~$31 \cdot 30 + 30 = 960 \le 999$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x \cdot x \le n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x=316$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x^2 \le 100000.$~$ |
---|

Fractional digits | $~$\log_b(316) \approx \frac{5\log_b(10)}{2}$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x \cdot x = n,$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$y$~$ |
---|

Fractional digits | $~$y \cdot y \cdot y = 216,$~$ |
---|

Fractional digits | $~$y$~$ |
---|

Fractional digits | $~$y = \sqrt[3]{2 \cdot 12 \cdot 9} = 6$~$ |
---|

Fractional digits | $~$\sqrt[2]{1 \cdot 10} \approx 3.16.$~$ |
---|

Fractional digits | $~$\sqrt[2]{10}$~$ |
---|

Fractional digits | $~$\sqrt[3]{1 \cdot 1 \cdot 10} \approx 2.15.$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$1 < n \le 10$~$ |
---|

Fractional digits | $~$\log_{3.16}(5.62) \approx 1.5$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$\sqrt{n}$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x \cdot x$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$\sqrt{n}$~$ |
---|

Fractional digits | $~$10^2 = 100.$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$n^2$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$\sqrt{n}$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$x > 1.$~$ |
---|

Fractional digits | $~$\sqrt[n]{10} > 1$~$ |
---|

Fractional digits | $~$n$~$ |
---|

Fractional digits | $~$x$~$ |
---|

Fractional digits | $~$0 < x < 1,$~$ |
---|

Free group | $~$F(X)$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$F(X)$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$F(X)$~$ |
---|

Free group | $~$FX$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X = \{ a, b \}$~$ |
---|

Free group | $~$(a,b,a,a,a,b^{-1})$~$ |
---|

Free group | $~$abaaab^{-1}$~$ |
---|

Free group | $~$aba^3b^{-1}$~$ |
---|

Free group | $~$()$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$(b,b,b)$~$ |
---|

Free group | $~$b^3$~$ |
---|

Free group | $~$(a^{-1}, b^{-1}, b^{-1})$~$ |
---|

Free group | $~$a^{-1} b^{-2}$~$ |
---|

Free group | $~$aa^{-1}$~$ |
---|

Free group | $~$c$~$ |
---|

Free group | $~$c$~$ |
---|

Free group | $~$\{a,b\}$~$ |
---|

Free group | $~$abb^{-1}a$~$ |
---|

Free group | $~$\cdot$~$ |
---|

Free group | $~$aba \cdot bab = ababab$~$ |
---|

Free group | $~$aba^2 \cdot a^3b = aba^5b$~$ |
---|

Free group | $~$aba^{-1} \cdot a = ab$~$ |
---|

Free group | $~$aba^{-1}a$~$ |
---|

Free group | $~$ab \cdot b^{-1} a^{-1} = \varepsilon$~$ |
---|

Free group | $~$abb^{-1}a^{-1} = aa^{-1}$~$ |
---|

Free group | $~$b$~$ |
---|

Free group | $~$a a^{-1} = \varepsilon$~$ |
---|

Free group | $~$\{ a \}$~$ |
---|

Free group | $~$a^n$~$ |
---|

Free group | $~$a^{-n}$~$ |
---|

Free group | $~$a^0$~$ |
---|

Free group | $~$a^i$~$ |
---|

Free group | $~$i \in \mathbb{Z}$~$ |
---|

Free group | $~$a^{i_1} b^{j_1} a^{i_2} b^{j_2} \dots a^{i_n} b^{j_n}$~$ |
---|

Free group | $$~$a^{i_1} b^{j_1} a^{i_2} b^{j_2} \dots a^{i_n} b^{j_n} \mapsto 2^{\mathrm{sgn}(i_1)+2} 3^{|i_1|} 5^{\mathrm{sgn}(j_1)+2} 7^{|j_1|} \dots$~$$ |
---|

Free group | $~$\mathrm{sgn}$~$ |
---|

Free group | $~$-1$~$ |
---|

Free group | $~$1$~$ |
---|

Free group | $~$0$~$ |
---|

Free group | $~$0$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$X = \{ a, b \}$~$ |
---|

Free group | $~$C_2$~$ |
---|

Free group | $~$a$~$ |
---|

Free group | $~$b \cdot b = a$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$b^2 = a$~$ |
---|

Free group | $~$FX$~$ |
---|

Free group | $~$a, b$~$ |
---|

Free group | $~$a$~$ |
---|

Free group | $~$b$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$a \cdot b$~$ |
---|

Free group | $~$a$~$ |
---|

Free group | $~$b$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$a \cdot b$~$ |
---|

Free group | $~$ab$~$ |
---|

Free group | $~$a^{-1} \cdot a$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$a^{-1} a$~$ |
---|

Free group | $~$a^{-1}ba^2b^{-2}$~$ |
---|

Free group | $~$G$~$ |
---|

Free group | $~$\langle X \mid R \rangle$~$ |
---|

Free group | $~$G$~$ |
---|

Free group | $~$F(X)$~$ |
---|

Free group | $~$F(X)$~$ |
---|

Free group | $~$G$~$ |
---|

Free group | $~$FX$~$ |
---|

Free group | $~$FY$~$ |
---|

Free group | $~$X$~$ |
---|

Free group | $~$Y$~$ |
---|

Free group | $~$\mathbb{Z}$~$ |
---|

Free group | $~$a, b$~$ |
---|

Free group | $~$ab \not = ba$~$ |
---|

Free group | $~$\rho_a \rho_b \not = \rho_b \rho_a$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$ab$~$ |
---|

Free group | $~$ba$~$ |
---|

Free group | $~$\varepsilon$~$ |
---|

Free group | $~$x \in \mathbb{Q}$~$ |
---|

Free group | $~$n \not = 0$~$ |
---|

Free group | $~$x+x+\dots+x$~$ |
---|

Free group | $~$n$~$ |
---|

Free group | $~$0$~$ |
---|

Free group | $~$(\mathbb{Q}, +)$~$ |
---|

Free group | $~$n \times x = 0$~$ |
---|

Free group | $~$n=0$~$ |
---|

Free group | $~$x = 0$~$ |
---|

Free group | $~$n \not = 0$~$ |
---|

Free group | $~$x = 0$~$ |
---|

Free group | $~$x$~$ |
---|

Free group | $~$\mathbb{Q}$~$ |
---|

Free group | $~$\mathbb{Z}$~$ |
---|

Free group | $~$\mathbb{Z}$~$ |
---|

Free group | $~$\mathbb{Z}$~$ |
---|

Free group | $~$1$~$ |
---|

Free group | $~$\mathbb{Z}$~$ |
---|

Free group | $~$1$~$ |
---|

Free group | $~$\mathbb{Q}$~$ |
---|

Free group | $~$x$~$ |
---|

Free group | $~$\frac{x}{2}$~$ |
---|

Free group | $~$x$~$ |
---|

Free group universal property | $~$X$~$ |
---|

Free group universal property | $~$FX$~$ |
---|

Free group universal property | $~$X$~$ |
---|

Free group universal property | $~$G$~$ |
---|

Free group universal property | $~$f: X \to G$~$ |
---|

Free group universal property | $~$G$~$ |
---|

Free group universal property | $~$G$~$ |
---|

Free group universal property | $~$\overline{f}: FX \to G$~$ |
---|

Free group universal property | $~$\overline{f}(\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}) = f(a_1) \cdot f(a_2) \cdot \dots \cdot f(a_n)$~$ |
---|

Free group universal property | $~$FX$~$ |
---|

Free group universal property | $~$G$~$ |
---|

Free group universal property | $~$f: X \to G$~$ |
---|

Free group universal property | $~$FX \to G$~$ |
---|

Free group universal property | $~$X$~$ |
---|

Free group universal property | $~$f$~$ |
---|

Free group universal property | $~$FX$~$ |
---|

Free group universal property | $~$FX$~$ |
---|

Free group universal property | $~$C_3$~$ |
---|

Free group universal property | $~$\{ e, a, b\}$~$ |
---|

Free group universal property | $~$e$~$ |
---|

Free group universal property | $~$a + a = b$~$ |
---|

Free group universal property | $~$a+b = e = b+a$~$ |
---|

Free group universal property | $~$b+b = a$~$ |
---|

Free group universal property | $~$a$~$ |
---|

Free group universal property | $~$a=a$~$ |
---|

Free group universal property | $~$a+a = b$~$ |
---|

Free group universal property | $~$a+a+a = e$~$ |
---|

Free group universal property | $~$G = (\mathbb{Z}, +)$~$ |
---|

Free group universal property | $~$f: C_3 \to \mathbb{Z}$~$ |
---|

Free group universal property | $~$a \mapsto 1$~$ |
---|

Free group universal property | $~$C_3$~$ |
---|

Free group universal property | $~$\{ e, a, b\}$~$ |
---|

Free group universal property | $~$\overline{f}: C_3 \to \mathbb{Z}$~$ |
---|

Free group universal property | $~$\overline{f}(a) = 1$~$ |
---|

Free group universal property | $~$f$~$ |
---|

Free group universal property | $~$\overline{f}$~$ |
---|

Free group universal property | $~$\overline{f}(e) = \overline{f}(a+a+a) = 1+1+1 = 3$~$ |
---|

Free group universal property | $~$\overline{f}(e) = 3$~$ |
---|

Free group universal property | $~$C_3$~$ |
---|

Free group universal property | $~$a+a+a = e$~$ |
---|

Free group universal property | $~$\overline{f}$~$ |
---|

Free group universal property | $~$C_3$~$ |
---|

Free groups are torsion-free | $~$FX$~$ |
---|

Free groups are torsion-free | $~$X$~$ |
---|

Free groups are torsion-free | $~$FX$~$ |
---|

Free groups are torsion-free | $~$X$~$ |
---|

Free groups are torsion-free | $~$a_1 a_2 \dots a_n$~$ |
---|

Free groups are torsion-free | $~$a_1 \not = a_n^{-1}$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$r w^\prime r^{-1}$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$w^\prime$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$r^{-1}$~$ |
---|

Free groups are torsion-free | $~$w^\prime$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$r = \varepsilon$~$ |
---|

Free groups are torsion-free | $~$w^\prime = w$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$a v a^{-1}$~$ |
---|

Free groups are torsion-free | $~$a \in X$~$ |
---|

Free groups are torsion-free | $~$v$~$ |
---|

Free groups are torsion-free | $~$v$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$v$~$ |
---|

Free groups are torsion-free | $~$r v^\prime r^{-1}$~$ |
---|

Free groups are torsion-free | $~$v^\prime$~$ |
---|

Free groups are torsion-free | $~$w = a r v^\prime r^{-1} a^{-1} = (ar) v^\prime (ar)^{-1}$~$ |
---|

Free groups are torsion-free | $~$r w^\prime r^{-1} = s v^\prime s^{-1}$~$ |
---|

Free groups are torsion-free | $~$s^{-1} r w^\prime r^{-1} s = v^\prime$~$ |
---|

Free groups are torsion-free | $~$v^\prime$~$ |
---|

Free groups are torsion-free | $~$s$~$ |
---|

Free groups are torsion-free | $~$v^\prime = r w^\prime r^{-1}$~$ |
---|

Free groups are torsion-free | $~$w = r w^\prime r^{-1}$~$ |
---|

Free groups are torsion-free | $~$r = e$~$ |
---|

Free groups are torsion-free | $~$v^\prime = w^\prime = w$~$ |
---|

Free groups are torsion-free | $~$s$~$ |
---|

Free groups are torsion-free | $~$r^{-1}$~$ |
---|

Free groups are torsion-free | $~$s$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$s$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$v^\prime = w^\prime$~$ |
---|

Free groups are torsion-free | $~$w$~$ |
---|

Free groups are torsion-free | $~$n$~$ |
---|

Free groups are torsion-free | $~$r w^\prime r^{-1}$~$ |
---|

Free groups are torsion-free | $~$(rw^\prime r^{-1})^n = r (w^\prime)^n r^{-1}$~$ |
---|

Free groups are torsion-free | $~$r$~$ |
---|

Free groups are torsion-free | $~$w^\prime$~$ |
---|

Free groups are torsion-free | $~$r^{-1}$~$ |
---|

Free groups are torsion-free | $~$r, (w^\prime)^n, r^{-1}$~$ |
---|

Free groups are torsion-free | $~$w^\prime$~$ |
---|

Free groups are torsion-free | $~$r (w^\prime)^n r^{-1}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$x x^{-1}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$x^{-1}$~$ |
---|

Freely reduced word | $$~$X^{-1} = \{ x^{-1} \mid x \in X \}$~$$ |
---|

Freely reduced word | $~$x^{-1}$~$ |
---|

Freely reduced word | $~$X \cup X^{-1}$~$ |
---|

Freely reduced word | $~$X \cup X^{-1}$~$ |
---|

Freely reduced word | $~$X \cup X^{-1}$~$ |
---|

Freely reduced word | $~$X = \{ 1, 2 \}$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$\varepsilon$~$ |
---|

Freely reduced word | $~$(1)$~$ |
---|

Freely reduced word | $~$(2)$~$ |
---|

Freely reduced word | $~$(2^{-1})$~$ |
---|

Freely reduced word | $~$(1, 2^{-1}, 2, 1, 1, 1, 2^{-1}, 1^{-1}, 1^{-1})$~$ |
---|

Freely reduced word | $~$\varepsilon$~$ |
---|

Freely reduced word | $~$1$~$ |
---|

Freely reduced word | $~$2$~$ |
---|

Freely reduced word | $~$2^{-1}$~$ |
---|

Freely reduced word | $~$1 2^{-1} 2 1 1 1 2^{-1} 1^{-1} 1^{-1}$~$ |
---|

Freely reduced word | $~$1 2^{-1} 2 1^3 2^{-1} 1^{-2}$~$ |
---|

Freely reduced word | $~$r r^{-1}$~$ |
---|

Freely reduced word | $~$r^{-1} r$~$ |
---|

Freely reduced word | $~$r \in X$~$ |
---|

Freely reduced word | $~$X = \{ a, b, c \}$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$\{ a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|

Freely reduced word | $~$\{ x, y, z \}$~$ |
---|

Freely reduced word | $~$a^{-1}$~$ |
---|

Freely reduced word | $~$x$~$ |
---|

Freely reduced word | $~$X \cup X^{-1} = \{ a,b,c, a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|

Freely reduced word | $~$X \cup X^{-1}$~$ |
---|

Freely reduced word | $~$\varepsilon$~$ |
---|

Freely reduced word | $~$a$~$ |
---|

Freely reduced word | $~$aaaa$~$ |
---|

Freely reduced word | $~$b$~$ |
---|

Freely reduced word | $~$b^{-1}$~$ |
---|

Freely reduced word | $~$ab$~$ |
---|

Freely reduced word | $~$ab^{-1}cbb^{-1}c^{-1}$~$ |
---|

Freely reduced word | $~$aa^{-1}aa^{-1}$~$ |
---|

Freely reduced word | $~$ab^{-1}cbb^{-1}c^{-1}$~$ |
---|

Freely reduced word | $~$bb^{-1}$~$ |
---|

Freely reduced word | $~$aa^{-1}aa^{-1}$~$ |
---|

Freely reduced word | $~$aa^{-1}$~$ |
---|

Freely reduced word | $~$a^{-1} a$~$ |
---|

Freely reduced word | $~$a^{-1}$~$ |
---|

Freely reduced word | $~$b^{-1}$~$ |
---|

Freely reduced word | $~$X^{-1}$~$ |
---|

Freely reduced word | $~$\{ x, y, z \}$~$ |
---|

Freely reduced word | $~$\{ a, b, c \}$~$ |
---|

Freely reduced word | $~$\{ a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|

Freely reduced word | $~$\varepsilon$~$ |
---|

Freely reduced word | $~$a$~$ |
---|

Freely reduced word | $~$aaaa$~$ |
---|

Freely reduced word | $~$a^4$~$ |
---|

Freely reduced word | $~$b$~$ |
---|

Freely reduced word | $~$y$~$ |
---|

Freely reduced word | $~$ab$~$ |
---|

Freely reduced word | $~$aycbyz$~$ |
---|

Freely reduced word | $~$axax$~$ |
---|

Freely reduced word | $~$aycbyz$~$ |
---|

Freely reduced word | $~$by$~$ |
---|

Freely reduced word | $~$axax$~$ |
---|

Freely reduced word | $~$ax$~$ |
---|

Freely reduced word | $~$xa$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X \cup X^{-1}$~$ |
---|

Freely reduced word | $~$r r^{-1}$~$ |
---|

Freely reduced word | $~$r^{-1} r$~$ |
---|

Freely reduced word | $~$r r^{-1}$~$ |
---|

Freely reduced word | $~$r \in X$~$ |
---|

Freely reduced word | $~$r^{-1} r$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Freely reduced word | $~$X$~$ |
---|

Function | $~$f$~$ |
---|

Function | $~$f$~$ |
---|

Function | $~$X$~$ |
---|

Function | $~$Y$~$ |
---|

Function | $~$-$~$ |
---|

Function | $~$(4, 3)$~$ |
---|

Function | $~$1,$~$ |
---|

Function | $~$(19, 2)$~$ |
---|

Function | $~$17,$~$ |
---|

Function | $~$f : X \to Y$~$ |
---|

Function | $~$f$~$ |
---|

Function | $~$X$~$ |
---|

Function | $~$Y$~$ |
---|

Function | $~$f$~$ |
---|

Function | $~$X$~$ |
---|

Function | $~$Y$~$ |
---|

Function | $~$- : (\mathbb N \times \mathbb N) \to \mathbb N,$~$ |
---|

Function | $~$X$~$ |
---|

Function | $~$f.$~$ |
---|

Function | $~$Y$~$ |
---|

Function | $~$f$~$ |
---|

Function | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|

Function | $~$f(x) = x^2$~$ |
---|

Function: Physical metaphor | $~$+$~$ |
---|

Function: Physical metaphor | $~$+$~$ |
---|

Function: Physical metaphor | $~$\times$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$3 \times 5$~$ |
---|

Fundamental Theorem of Arithmetic | $~$3 \times 5 \times 1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$15$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|

Fundamental Theorem of Arithmetic | $~$0$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$17 \times 23 \times 23$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$17 \times 23^2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\{ 17, 23, 23\}$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$a \times b$~$ |
---|

Fundamental Theorem of Arithmetic | $~$a$~$ |
---|

Fundamental Theorem of Arithmetic | $~$b$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$a$~$ |
---|

Fundamental Theorem of Arithmetic | $~$b$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$a$~$ |
---|

Fundamental Theorem of Arithmetic | $~$b$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n = 1274$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$49 \times 26$~$ |
---|

Fundamental Theorem of Arithmetic | $~$49$~$ |
---|

Fundamental Theorem of Arithmetic | $~$7^2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$26$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2 \times 13$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2 \times 7^2 \times 13$~$ |
---|

Fundamental Theorem of Arithmetic | $~$49$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|

Fundamental Theorem of Arithmetic | $~$26$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p$~$ |
---|

Fundamental Theorem of Arithmetic | $~$ab$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p$~$ |
---|

Fundamental Theorem of Arithmetic | $~$a$~$ |
---|

Fundamental Theorem of Arithmetic | $~$b$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n = 2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1 p_2 \dots p_r$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1 q_2 \dots q_s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_j$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1 = p_2 = q_3 = q_7$~$ |
---|

Fundamental Theorem of Arithmetic | $~$r=s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_i = q_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$n$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1 p_2 \dots p_r$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1 q_2 \dots q_s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_2 \dots q_s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_3 \dots q_s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$i=1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1 = q_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_2 \dots p_r = q_2 \dots q_s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$r-1 = s-1$~$ |
---|

Fundamental Theorem of Arithmetic | $~$r=s$~$ |
---|

Fundamental Theorem of Arithmetic | $~$p_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|

Fundamental Theorem of Arithmetic | $~$i \geq 2$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\mathbb{Z}[\sqrt{-5}]$~$ |
---|

Fundamental Theorem of Arithmetic | $~$\mathbb{Z}[\sqrt{-3}]$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$f : X \times X \to X$~$ |
---|

Generalized associative law | $~$X$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$[a, b, c, d, e],$~$ |
---|

Generalized associative law | $~$a \cdot b$~$ |
---|

Generalized associative law | $~$ab.$~$ |
---|

Generalized associative law | $~$((ab)c)(de)$~$ |
---|

Generalized associative law | $~$a$~$ |
---|

Generalized associative law | $~$b$~$ |
---|

Generalized associative law | $~$c$~$ |
---|

Generalized associative law | $~$d$~$ |
---|

Generalized associative law | $~$e$~$ |
---|

Generalized associative law | $~$a(b(c(de))$~$ |
---|

Generalized associative law | $~$d$~$ |
---|

Generalized associative law | $~$e$~$ |
---|

Generalized associative law | $~$c$~$ |
---|

Generalized associative law | $~$b$~$ |
---|

Generalized associative law | $~$a$~$ |
---|

Generalized associative law | $~$abcde$~$ |
---|

Generalized associative law | $~$[a, b, c, d, e]$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f_4$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f_5$~$ |
---|

Generalized associative law | $~$f,$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$(x\cdot y) \cdot z = x \cdot (y \cdot z).$~$ |
---|

Generalized associative law | $~$x \cdot y$~$ |
---|

Generalized associative law | $~$xy,$~$ |
---|

Generalized associative law | $~$[a, b, c, d]$~$ |
---|

Generalized associative law | $~$a(b(cd)),$~$ |
---|

Generalized associative law | $~$\cdot$~$ |
---|

Generalized associative law | $~$a(b(cd))=a((bc)d)=(a(bc))d=((ab)c)d=(ab)(cd).$~$ |
---|

Generalized associative law | $~$f : X \times X \to X$~$ |
---|

Generalized associative law | $~$f_n$~$ |
---|

Generalized associative law | $~$n$~$ |
---|

Generalized associative law | $~$n \ge 1$~$ |
---|

Generalized associative law | $~$f_1$~$ |
---|

Generalized associative law | $~$f,$~$ |
---|

Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|

Generalized associative law | $~$\alpha,$~$ |
---|

Generalized associative law | $~$[x, y, z, \ldots]$~$ |
---|

Generalized associative law | $~$\chi,$~$ |
---|

Generalized associative law | $~$f(\alpha, \chi)$~$ |
---|

Generalized associative law | $~$[a, b, c, \ldots, x, y, z, \ldots]:$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f$~$ |
---|

Generalized associative law | $~$f_n : X^n \to X$~$ |
---|

Generalized associative law | $~$n \ge 0,$~$ |
---|

Generalized associative law | $~$0_X$~$ |
---|

Generalized associative law | $~$X$~$ |
---|

Generalized associative law | $~$f_0$~$ |
---|

Generalized associative law | $~$0_X$~$ |
---|

Generalized associative law | $~$f.$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$x : A \to X$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$x$~$ |
---|

Generalized element | $~$I$~$ |
---|

Generalized element | $~$*$~$ |
---|

Generalized element | $~$I = \{*\}$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$I$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$x$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$I$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$f(i) = x$~$ |
---|

Generalized element | $~$i \in I$~$ |
---|

Generalized element | $~$f$~$ |
---|

Generalized element | $~$x$~$ |
---|

Generalized element | $~$f : I \to X$~$ |
---|

Generalized element | $~$*$~$ |
---|

Generalized element | $~$I$~$ |
---|

Generalized element | $~$f(*)$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$I$~$ |
---|

Generalized element | $~$I \to X$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$n$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$n$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$1$~$ |
---|

Generalized element | $~$1$~$ |
---|

Generalized element | $~$\mathbb{Z}$~$ |
---|

Generalized element | $~$\mathbb{Z}$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$\text{Set} \times \text{Set}$~$ |
---|

Generalized element | $~$(X,Y)$~$ |
---|

Generalized element | $~$(2^A, 2^{X + B})$~$ |
---|

Generalized element | $~$(2^{Y + A}, 2^{B})$~$ |
---|

Generalized element | $~$(X,Y)$~$ |
---|

Generalized element | $~$(2^A)^X\times(2^{X+B})^Y \cong 2^{X\times A + Y \times (X + B)} \cong 2^{X \times A + Y \times B + X \times Y}$~$ |
---|

Generalized element | $~$(X,Y)$~$ |
---|

Generalized element | $~$(2^{Y+A})^X \times (2^B)^Y \cong 2^{X\times(Y+A) + Y \times B} \cong 2^{X \times A + Y \times B + X \times Y}$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$Y$~$ |
---|

Generalized element | $~$(0,1)$~$ |
---|

Generalized element | $~$(1,0)$~$ |
---|

Generalized element | $~$x$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$f$~$ |
---|

Generalized element | $~$X$~$ |
---|

Generalized element | $~$Y$~$ |
---|

Generalized element | $~$f(x) := f\circ x$~$ |
---|

Generalized element | $~$A$~$ |
---|

Generalized element | $~$Y$~$ |
---|

Generalized element | $~$f(xu) = f(x) u$~$ |
---|

Geometric product | $~$e^{\text{I}\theta}$~$ |
---|

Geometric product | $~$n$~$ |
---|

Geometric product | $~$|a|^2 + |b|^2 = |a+b|^2$~$ |
---|

Geometric product | $~$(a+b)^2 = a^2 + ab + ba + b^2$~$ |
---|

Geometric product | $~$ab$~$ |
---|

Geometric product | $~$ba$~$ |
---|

Geometric product | $~$2ab$~$ |
---|

Geometric product | $~$a^2 = |a|^2$~$ |
---|

Geometric product | $~$|a+b|^2 = |a|^2 + ab + ba + |b|^2$~$ |
---|

Geometric product | $~$ab + ba$~$ |
---|

Geometric product | $~$ab + ba$~$ |
---|

Geometric product | $~$a$~$ |
---|

Geometric product | $~$b$~$ |
---|

Geometric product | $~$a$~$ |
---|

Geometric product | $~$b$~$ |
---|

Geometric product | $~$|a+b|^2 = (|a| + |b|)^2 = |a|^2 + 2|a||b| + |b|^2$~$ |
---|

Geometric product | $~$ab + ba$~$ |
---|

Geometric product | $~$2|a||b|$~$ |
---|

Geometric product | $~$ab = - ba$~$ |
---|

Geometric product | $~$ab = ba = |a||b|$~$ |
---|

Geometric product | $~$\frac{1}{a}=\frac{a}{|a|^2}$~$ |
---|

Geometric product | $~$a^{-1}$~$ |
---|

Geometric product | $~$ae^{\text{I}\pi/2} = b$~$ |
---|

Geometric product | $~$e^{\text{I}\pi/2} = \frac{ab}{|a|^2}$~$ |
---|

Geometric product | $~$ab = |a|^2e^{\text{I}\pi/2}$~$ |
---|

Geometric product | $~$|a| = |b|$~$ |
---|

Geometric product | $~$b$~$ |
---|

Geometric product | $~$a|b|/|a|e^{\text{I}\pi/2} = b$~$ |
---|

Geometric product | $~$ab = |a||b|e^{\text{I}\pi/2}$~$ |
---|

Geometric product | $~$|b||a|=-e^{\text{I}\pi/2}$~$ |
---|

Geometric product | $~$e^{\text{I}\pi/2}$~$ |
---|

Geometric product | $~$\text{I}$~$ |
---|

Geometric product | $~$ab = |a||b|I$~$ |
---|

Geometric product | $~$I^2 = -1$~$ |
---|

Geometric product | $~$ab$~$ |
---|

Geometric product | $~$a = a_xx+a_yy$~$ |
---|

Geometric product | $~$b = b_xx+b_yy$~$ |
---|

Geometric product | $~$ab = (a_xx + a_yy)(b_xx + b_yy) = a_xb_xx^2 + a_yb_xyx + a_xb_yxy+a_yb_yy^2 = a_xb_x + a_yb_y - a_yb_xI + a_xb_yI$~$ |
---|

Geometric product | $~$e^{\text{I}\pi/4} = \frac{1 + I}{\sqrt{2}}$~$ |
---|

Geometric product | $~$e^{\text{I}\theta} = \cos(\theta) + \text{I}\sin(\theta)$~$ |
---|

Geometric product | $~$k$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf x$~$ |
---|

Geometry of vectors: direction | $~$\mathbf y$~$ |
---|

Geometry of vectors: direction | $~$\mathbf z$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf B$~$ |
---|

Geometry of vectors: direction | $~$\mathbf I$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf {x},\mathbf{y})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{y},\mathbf{z})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{z})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf {x},\mathbf{y})$~$ |
---|

Geometry of vectors: direction | $~$\mathbf w$~$ |
---|

Geometry of vectors: direction | $~$\mathbf x$~$ |
---|

Geometry of vectors: direction | $~$\mathbf y$~$ |
---|

Geometry of vectors: direction | $~$\mathbf z$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{w},\mathbf{x})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{w},\mathbf{y})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf {w},\mathbf{z})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{y})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{z})$~$ |
---|

Geometry of vectors: direction | $~$(\mathbf{y},\mathbf{z})$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf b$~$ |
---|

Geometry of vectors: direction | $~$\mathbf a$~$ |
---|

Geometry of vectors: direction | $~$\pi$~$ |
---|

Geometry of vectors: direction | $~$3.14$~$ |
---|

Geometry of vectors: direction | $~$\pi$~$ |
---|

Geometry of vectors: direction | $~$\frac{\pi}{2}$~$ |
---|

Geometry of vectors: direction | $~$\frac{\pi}{2}$~$ |
---|

Geometry of vectors: direction | $~$0$~$ |
---|

Geometry of vectors: direction | $~$\pi$~$ |
---|

Geometry of vectors: direction | $~$\frac{\pi}{4}$~$ |
---|

Geometry of vectors: direction | $~$R$~$ |
---|

Geometry of vectors: direction | $~$\mathbf B$~$ |
---|

Geometry of vectors: direction | $~$e$~$ |
---|

Geometry of vectors: direction | $~$R = e^{\mathbf B}$~$ |
---|

Goodhart's Curse | $~$V$~$ |
---|

Goodhart's Curse | $~$V$~$ |
---|

Goodhart's Curse | $~$U$~$ |
---|

Goodhart's Curse | $~$V,$~$ |
---|

Goodhart's Curse | $~$U$~$ |
---|

Goodhart's Curse | $~$V,$~$ |
---|

Goodhart's Curse | $~$U$~$ |
---|

Goodhart's Curse | $~$U$~$ |
---|

Goodhart's Curse | $~$V.$~$ |
---|

Goodhart's Curse | $~$U$~$ |
---|

Goodhart's Curse | $~$U-V$~$ |
---|

Goodhart's Curse | $~$\|U - V\|$~$ |
---|

Graham's number | $~$f(x) = 3\uparrow^n 3$~$ |
---|

Graham's number | $~$f^n(x) = \underbrace{f(f(f(\cdots f(f(x)) \cdots ))}_{n\text{ applications of }f}$~$ |
---|

Graham's number | $~$f^{64}(4).$~$ |
---|

Greatest common divisor | $~$a$~$ |
---|

Greatest common divisor | $~$b$~$ |
---|

Greatest common divisor | $~$a$~$ |
---|

Greatest common divisor | $~$b$~$ |
---|

Greatest common divisor | $~$a$~$ |
---|

Greatest common divisor | $~$b$~$ |
---|

Greatest common divisor | $~$c$~$ |
---|

Greatest common divisor | $~$c \mid a$~$ |
---|

Greatest common divisor | $~$c \mid b$~$ |
---|

Greatest common divisor | $~$d \mid a$~$ |
---|

Greatest common divisor | $~$d \mid b$~$ |
---|

Greatest common divisor | $~$d \mid c$~$ |
---|

Greatest common divisor | $~$a$~$ |
---|

Greatest common divisor | $~$b$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$P$~$ |
---|

Greatest lower bound in a poset | $~$\leq$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$P$~$ |
---|

Greatest lower bound in a poset | $~$z \in P$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$z \leq x$~$ |
---|

Greatest lower bound in a poset | $~$z \leq y$~$ |
---|

Greatest lower bound in a poset | $~$z \in P$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$z$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$w$~$ |
---|

Greatest lower bound in a poset | $~$x$~$ |
---|

Greatest lower bound in a poset | $~$y$~$ |
---|

Greatest lower bound in a poset | $~$w \leq z$~$ |
---|

Group | $~$120^\circ$~$ |
---|

Group | $~$240^\circ$~$ |
---|

Group | $~$f$~$ |
---|

Group | $~$g$~$ |
---|

Group | $~$h$~$ |
---|

Group | $~$g \circ f$~$ |
---|

Group | $~$h \circ (g \circ f)$~$ |
---|

Group | $~$h \circ g$~$ |
---|

Group | $~$(h \circ g) \circ f$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$(X, \bullet)$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$x, y$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$xy$~$ |
---|

Group | $~$x(yz) = (xy)z$~$ |
---|

Group | $~$x, y, z \in X$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$xe=ex=x$~$ |
---|

Group | $~$x \in X$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x^{-1} \in X$~$ |
---|

Group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|

Group | $~$120^\circ$~$ |
---|

Group | $~$240^\circ$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$(X, \bullet)$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$\bullet : G \times G \to G$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$xy$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$x, y$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$xy$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$xe=ex=x$~$ |
---|

Group | $~$x \in X$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x^{-1} \in X$~$ |
---|

Group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|

Group | $~$x(yz) = (xy)z$~$ |
---|

Group | $~$x, y, z \in X$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$G \times G \to G$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$z$~$ |
---|

Group | $~$ze = ez = z.$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$e$~$ |
---|

Group | $~$1$~$ |
---|

Group | $~$1_G$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$1$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$0$~$ |
---|

Group | $~$0_G$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$y$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$xy = e$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$x^{-1}$~$ |
---|

Group | $~$(-x)$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$f$~$ |
---|

Group | $~$g$~$ |
---|

Group | $~$h$~$ |
---|

Group | $~$g \circ f$~$ |
---|

Group | $~$h \circ (g \circ f)$~$ |
---|

Group | $~$h \circ g$~$ |
---|

Group | $~$(h \circ g) \circ f$~$ |
---|

Group | $~$(\mathbb{Z}, +)$~$ |
---|

Group | $~$\mathbb{Z}$~$ |
---|

Group | $~$+$~$ |
---|

Group | $~$\mathbb Z \times \mathbb Z \to \mathbb Z$~$ |
---|

Group | $~$(x+y)+z=x+(y+z)$~$ |
---|

Group | $~$0+x = x = x + 0$~$ |
---|

Group | $~$x$~$ |
---|

Group | $~$-x$~$ |
---|

Group | $~$x + (-x) = 0$~$ |
---|

Group | $~$G = (X, \bullet)$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$\bullet$~$ |
---|

Group | $~$x \bullet y$~$ |
---|

Group | $~$xy$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$x, y \in X$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$x, y \in G$~$ |
---|

Group | $~$G$~$ |
---|

Group | $~$|G|$~$ |
---|

Group | $~$|X|$~$ |
---|

Group | $~$X$~$ |
---|

Group | $~$|G|=9$~$ |
---|

Group | $~$G$~$ |
---|

Group action | $~$G$~$ |
---|

Group action | $~$X$~$ |
---|

Group action | $~$\alpha : G \times X \to X$~$ |
---|

Group action | $~$(g, x) \mapsto gx$~$ |
---|

Group action | $~$\alpha$~$ |
---|

Group action | $~$ex = x$~$ |
---|

Group action | $~$x \in X$~$ |
---|

Group action | $~$e$~$ |
---|

Group action | $~$g(hx) = (gh)x$~$ |
---|

Group action | $~$g, h \in G, x \in X$~$ |
---|

Group action | $~$gh$~$ |
---|

Group action | $~$G$~$ |
---|

Group action | $~$G$~$ |
---|

Group action | $~$X$~$ |
---|

Group action | $~$G \to \text{Aut}(X)$~$ |
---|

Group action | $~$\text{Aut}(X)$~$ |
---|

Group action | $~$X$~$ |
---|

Group action | $~$X \to X$~$ |
---|

Group action | $~$X = \mathbb{R}^2$~$ |
---|

Group action | $~$\mathbb{R}^2$~$ |
---|

Group action | $~$ISO(2)$~$ |
---|

Group action | $~$f : \mathbb{R}^2 \to \mathbb{R}^2$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho: G \times X \to X$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$G$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$X$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$X \to X$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$x \mapsto \rho(g, x)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})(\rho(g)(x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})(\rho(g, x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1}, \rho(g, x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1} g, x) = \rho(e, x) = x$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$e$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(g^{-1})(x)) = x$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$G$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$G \times X$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$X$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho: G \to \mathrm{Sym}(X)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(gh) = \rho(g) \rho(h)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(gh)(x) = \rho(gh, x)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(gh)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g, \rho(h, x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(h, x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(h)(x))$~$ |
---|

Group action induces homomorphism to the symmetric group | $~$\rho(h)$~$ |
---|

Group conjugate | $~$x, y$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$h \in G$~$ |
---|

Group conjugate | $~$hxh^{-1} = y$~$ |
---|

Group conjugate | $~$h$~$ |
---|

Group conjugate | $~$h$~$ |
---|

Group conjugate | $$~$\sigma = (a_{11} a_{12} \dots a_{1 n_1})(a_{21} \dots a_{2 n_2}) \dots (a_{k 1} a_{k 2} \dots a_{k n_k})$~$$ |
---|

Group conjugate | $~$\tau \in S_n$~$ |
---|

Group conjugate | $$~$\tau \sigma \tau^{-1} = (\tau(a_{11}) \tau(a_{12}) \dots \tau(a_{1 n_1}))(\tau(a_{21}) \dots \tau(a_{2 n_2})) \dots (\tau(a_{k 1}) \tau(a_{k 2}) \dots \tau(a_{k n_k}))$~$$ |
---|

Group conjugate | $~$\tau$~$ |
---|

Group conjugate | $~$\sigma$~$ |
---|

Group conjugate | $~$\tau$~$ |
---|

Group conjugate | $~$D_{2n}$~$ |
---|

Group conjugate | $~$n$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$X$~$ |
---|

Group conjugate | $~$g \in G$~$ |
---|

Group conjugate | $~$h \in G$~$ |
---|

Group conjugate | $~$hgh^{-1}$~$ |
---|

Group conjugate | $~$g$~$ |
---|

Group conjugate | $~$X$~$ |
---|

Group conjugate | $~$h$~$ |
---|

Group conjugate | $~$H$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$H$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$\rho: G \times G \to G$~$ |
---|

Group conjugate | $~$\rho(g, k) = g k g^{-1}$~$ |
---|

Group conjugate | $~$\rho(gh, k) = (gh)k(gh)^{-1} = ghkh^{-1}g^{-1} = g \rho(h, k) g^{-1} = \rho(g, \rho(h, k))$~$ |
---|

Group conjugate | $~$\rho(e, k) = eke^{-1} = k$~$ |
---|

Group conjugate | $~$\mathrm{Stab}_G(g)$~$ |
---|

Group conjugate | $~$g \in G$~$ |
---|

Group conjugate | $~$kgk^{-1} = g$~$ |
---|

Group conjugate | $~$kg = gk$~$ |
---|

Group conjugate | $~$g$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group conjugate | $~$\mathrm{Orb}_G(g)$~$ |
---|

Group conjugate | $~$g \in G$~$ |
---|

Group conjugate | $~$g$~$ |
---|

Group conjugate | $~$G$~$ |
---|

Group coset | $~$H$~$ |
---|

Group coset | $~$G$~$ |
---|

Group coset | $~$H$~$ |
---|

Group coset | $~$G$~$ |
---|

Group coset | $~$\{ gh : h \in H \}$~$ |
---|

Group coset | $~$g \in G$~$ |
---|

Group coset | $~$gH$~$ |
---|

Group coset | $~$Hg = \{ hg: h \in H \}$~$ |
---|

Group coset | $~$S_3$~$ |
---|

Group coset | $~$\{ e, (123), (132), (12), (13), (23) \}$~$ |
---|

Group coset | $~$A_3$~$ |
---|

Group coset | $~$\{ e, (123), (132) \}$~$ |
---|

Group coset | $~$(12) A_3$~$ |
---|

Group coset | $~$\{ (12), (12)(123), (12)(132) \}$~$ |
---|

Group coset | $~$\{ (12), (23), (13) \}$~$ |
---|

Group coset | $~$(123)A_3$~$ |
---|

Group coset | $~$A_3$~$ |
---|

Group coset | $~$A_3$~$ |
---|

Group coset | $~$(123)$~$ |
---|

Group coset | $~$A_3$~$ |
---|

Group coset | $~$H$~$ |
---|

Group coset | $~$G$~$ |
---|

Group coset | $~$G$~$ |
---|

Group coset | $~$H$~$ |
---|

Group coset | $~$H$~$ |
---|

Group coset | $~$G$~$ |
---|

Group coset | $~$p$~$ |
---|

Group coset | $~$p$~$ |
---|

Group homomorphism | $~$(G, +)$~$ |
---|

Group homomorphism | $~$(H, *)$~$ |
---|

Group homomorphism | $~$G$~$ |
---|

Group homomorphism | $~$H$~$ |
---|

Group homomorphism | $~$G$~$ |
---|

Group homomorphism | $~$H$~$ |
---|

Group homomorph |
---|