---
title: Chaos Background
author: Tristan Britt  (\texttt{tristan@warlock.xyz}), 0xAlcibiades (\texttt{alcibiades@warlock.xyz})
date: July 2024
---

## Glossary of Notation

| Notation | Meaning |
|--------|---------|
| $\{ \}$ | Set delimiters |
| $\varnothing$ | Empty set |
| $\in$ | Element of |
| $\notin$ | Not an element of |
| $\subset$ | Proper subset |
| $\subseteq$ | Subset or equal to |
| $\supset$ | Proper superset |
| $\supseteq$ | Superset or equal to |
| $\cup$ | Union |
| $\cap$ | Intersection |
| $\setminus$ | Set difference |
| $\overline{A}$ | Complement of set $A$ |
| $A^c$ | Complement of set $A$ (alternative notation) |
| $\mathcal{P}(A)$ | Power set of $A$ |
| $A \times B$ | Cartesian product of sets $A$ and $B$ |
| $|A|$ | Cardinality (size) of set $A$ |
| $\aleph_0$ | Cardinality of the natural numbers (countable infinity) |
| $\mathfrak{c}$ | Cardinality of the real numbers (continuum) |
| $\forall$ | For all |
| $\exists$ | There exists |
| $\exists!$ | There exists a unique |
| $:$ or $\mid$ | Such that |
| $\{ x \in A \mid P(x) \}$ | Set-builder notation: set of all $x$ in $A$ such that $P(x)$ is true |
| $[a,b]$ | Closed interval from $a$ to $b$ |
| $(a,b)$ | Open interval from $a$ to $b$ |
| $[a,b)$ or $(a,b]$ | Half-open intervals |
| $A \triangle B$ | Symmetric difference of sets $A$ and $B$ |
| $\bigsqcup$ | Disjoint union |
| $\bigcup_{i \in I} A_i$ | Union of a family of sets |
| $\bigcap_{i \in I} A_i$ | Intersection of a family of sets |
| $A^n$ | Cartesian product of $A$ with itself $n$ times |
| $f: A \to B$ | Function $f$ from set $A$ to set $B$ |
| $f(A)$ | Image of set $A$ under function $f$ |
| $f^{-1}(B)$ | Preimage of set $B$ under function $f$ |
| $\text{dom}(f)$ | Domain of function $f$ |
| $\text{cod}(f)$ | Codomain of function $f$ |
| $\text{range}(f)$ | Range of function $f$ |
| $\text{id}_A$ | Identity function on set $A$ |
| $f \circ g$ | Composition of functions $f$ and $g$ |
| $f|_A$ | Restriction of function $f$ to set $A$ |
| $f: A \twoheadrightarrow B$ | Surjective function from $A$ to $B$ |
| $f: A \hookrightarrow B$ | Injective function from $A$ to $B$ |
| $f: A \xrightarrow{\sim} B$ | Bijective function from $A$ to $B$ |
| $\mathbb{Z}$ | set of all integers |
| $\mathbb{Q}$ | set of all rational numbers |
| $\mathbb{R}$ | set of all real numbers |
| $\mathbb{C}$ | set of all complex numbers |
| $\Leftrightarrow$, iff | if and only if |
| $\mathbb{Z}^+, \mathbb{Q}^+, \mathbb{R}^+$ | sets of all positive integers, rational numbers, and real numbers, respectively |
| $a\mid b$ | $a$ divides $b$ |
| $*$ | binary operation |
| $\Delta$ | symmetric difference |
| $e$ | identity element of a group |
| $GL(2,\mathbb{R})$ | general linear group of degree 2 over $\mathbb{R}$ |
| $P(X)$ | set of subsets $X$ |
| $\mathbb{Z}_n$ | the set $\{0, 1, 2, \ldots, n-1\}$ |
| $a \equiv b \pmod{n}$ | the integers $a$ and $b$ are congruent modulo $n$ |
| $\oplus, \otimes$ | addition and multiplication modulo $n$ |
| $o(x)$ | order of the element $x$ |
| $\langle x \rangle$ | set of powers of the element $x$ |
| $\|G\|$ | order of the group $G$ |
| $V$ | Klein's 4-group |
| $Z(G)$ | center of the group $G$ |
| $GL(2,\mathbb{C})$ | general linear group of degree 2 over $\mathbb{C}$ |
| $Q_8$ | group of unit quaternions |
| $SL(2,\mathbb{R})$ | special linear group of degree 2 over $\mathbb{R}$ |
| $Z(g)$ | centralizer of the element $g$ |
| $G \times H$ | direct product of $G$ and $H$ |
| $f:S \to T$ | $f$ is a function from $S$ to $T$ |
| $f^{-1}$ | the inverse of the function $f$ |
| $g \circ f$ | composite function |
| $i_X$ | identity function on the set $X$ |
| $S_X$ | symmetric group on $X$ |
| $S_n$ | symmetric group of degree $n$ |
| $A_n$ | alternating group of degree $n$ |
| $D_4$ | group of symmetries of a square |
| $x \equiv_H y$ | means $xy^{-1} \in H$ |
| $x_H \equiv y$ | means $x^{-1}y \in H$ |
| $[G:H]$ | the index of $H$ in $G$ |
| $H \triangleleft G$ | $H$ is a normal subgroup of $G$ |
| $G/H$ | quotient group of $G$ by $H$ |
| $G \cong H$ | $G$ and $H$ are isomorphic |
| $\varphi^{-1}(J)$ | inverse image of $J$ under $\varphi$ |
| $\text{Aut}(G)$ | group of automorphisms of the group $G$ |
| $\rho$ | canonical homomorphism |
| $\ker(\varphi)$ | kernel of the homomorphism $\varphi$ |
| $N(H)$ | normalizer of the subgroup $H$ |
| $R \oplus S$ | direct sum of the rings $R$ and $S$ |
| $M_2(\mathbb{R})$ | ring of all $2 \times 2$ real matrices |
| $\mathbb{Z}[i]$ | ring of Gaussian integers |
| $\mathbb{H}$ | ring of quaternions |
| $R/I$ | quotient ring of $R$ by $I$ |
| $R[X]$ | polynomial ring over $R$ |
| $F(a)$ | field obtained by adjoining $a$ to the field $F$ |
| $\text{irr}(a/F)$ | irreducible polynomial of $a$ over $F$ |
| $\deg(a/F)$ | degree of $a$ over $F$ |
| $[E : F]$ | degree of the field $E$ over the field $F$ |
| $\mathbb{C}_c$ | field of constructible complex numbers |
| $\Gamma(E/F)$ | Galois group of $E$ over $F$ |
| $\Phi(H)$ | fixed field of the subgroup $H$ of $\Gamma(E/F)$ |
| $\Gamma(f(X)/F)$ | Galois group of $f(X)$ over $F$ |