# Debug - All Mathjax (18305)

pageMathJax
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..."

$C = AB; c_{ii} = a_{ii} * b_{ii}; ∀ i ≠ j, c_{ij} = 0$

$X_i$

$x_i$

$X_i$

$X_0$

$X_1$

$X_2$

$X_3$

$x_i$

"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..."

$x^2 \le 100000.$

$M$

$N$

"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 ..."

$x$

$n$

$2^n$

$2^{3,000,000,000,000}$

$2^{3,000,000,000,000}$

$2^\text{3 trillion}$

$a$

$b$

$31a + b$

$31\cdot 30 + 30 = 960$

"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..."

$\sqrt{10}$

$\log_{10}(\text{2,310,426})$

"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..."

$x$

$\log_{10}(12) \approx 1.08$

$\log_2(10) \approx 3.32$

"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..."

$\operatorname{d}\!f$

$X$

$\mathbb P(X)$

$X.$

"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..."

$f$

$\mathsf{Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {CooperateBot},$

$\mathsf {Fairbot}$

$\mathsf {CooperateBot},$

$\mathsf {Fairbot}$

"“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

$aa^{-1}b$

$(\exists v: \forall w > v: \forall x>0, y>0, z>0: x^w + y^w \neq z^w) \rightarrow ((1 = 0) \vee (1 + 0 = 0 + 1))$

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)

$n$

$\frac{1}{10} + \frac{1}{5}$

$$\frac{1}{10} + \frac{1}{5} = \frac{1 \times 5 + 10 \times 1}{10 \times 5} = \frac{5+10}{50} = \frac{15}{50}$$

$\frac{3}{10}$

$\frac{3}{10}$

$\frac{1}{10}$

$15$

$\frac{1}{50}$

$\frac{1}{10}$

$\frac{1}{5}$

$\frac{1}{5} = \frac{2}{10}$

$\frac{1}{10} + \frac{2}{10}$

$\frac{3}{10}$

$\frac{1}{15} + \frac{1}{10}$

$$\frac{1}{10} + \frac{1}{15} = \frac{1 \times 15 + 10 \times 1}{10 \times 15} = \frac{25}{150} = \frac{1}{6}$$

$\frac{1}{30}$

$\frac{1}{10}$

$\frac{1}{15}$

$\frac{3}{30} + \frac{2}{30} = \frac{5}{30}$

$\frac{5}{30} = \frac{1}{6}$

$\frac{25}{150} = \frac{1}{6}$

$\frac{1}{10} + \frac{1}{15}$

$$\frac{1}{15} + \frac{1}{10} = \frac{1 \times 10 + 15 \times 1}{15 \times 10} = \frac{25}{150} = \frac{1}{6}$$

$\frac{1}{10} + \frac{1}{15} = \frac{1}{15} + \frac{1}{10}$

$\frac{1}{6}$

$\frac{0}{5} + \frac{2}{5}$

$5$

$\frac{1}{5}$

$0$

$2$

$2$

$\frac{2}{5}$

$\frac{0}{7} + \frac{2}{5}$

$\frac{1}{7}$

$\frac{1}{5}$

$\frac{2}{5}$

$\frac{1}{7}$

$$\frac{0}{7} + \frac{2}{5} = \frac{0 \times 5 + 2 \times 7}{5 \times 7} = \frac{0 + 14}{35} = \frac{14}{35}$$

$\frac{2}{5}$

$\frac{1}{5}$

$\frac{1}{5} + \frac{-1}{10}$

$$\frac{1}{15} + \frac{-1}{10} = \frac{1 \times 10 + 15 \times (-1)}{15 \times 10} = \frac{10 - 15}{150} = \frac{-5}{150} = \frac{-1}{30}$$

$\frac{7}{8}$

$\frac{13}{8}$

$\frac{a}{b}$

$a$

$b$

$\frac{1}{8}$

$\frac{1}{8}$

$7$

$13$

$6$

$\frac{6}{8}$

$\frac{3}{4}$

$\frac{7}{8}$

$\frac{13}{7}$

$\frac{a}{b}$

$a$

$b$

$\frac{1}{8 \times 7} = \frac{1}{56}$

$\frac{1}{8}$

$\frac{1}{7}$

$\frac{7 \times 7}{7 \times 8} = \frac{49}{56}$

$\frac{8 \times 13}{8 \times 7} = \frac{104}{56}$

$49$

$104$

$55$

$\frac{1}{56}$

$\frac{55}{56}$

$\mathbb P(Y|X)$

$X$

$Y$

$Y,$

$X.$

$X$

$Y.$

$\pi_0$

$\mathbb E [U | \operatorname{do}(\pi_0), HumansObeyPlan]$

$\mathbb E [U | \operatorname{do}(\pi_0)],$

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

$P_\omega(A)$

$bayes_rule_details,$

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

$GL$

$\neg X$

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