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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