Notation

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The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.

Symbol	Description	Location
$x \in S$	$x$ is an element of $S$	Definition 1.1.2
$x \not\in S$	$x$ is not an element of $S$	Definition 1.1.2
$\emptyset$	the empty set, $\{\}$	Definition 1.1.2
$\mathbb{Z}$	the set of all integers	Example 1.1.1
$\mathbb{Q}$	the set of all rational numbers	Example 1.1.1
$\mathbb{R}$	the set of all real numbers	Example 1.1.1
$\mathbb{C}$	the set of all complex numbers	Example 1.1.1
$\mathbb{N}$	the set of all natural numbers, $\{0,1,2,\ldots\}$	Example 1.1.1
$\mathbb{Z}^+,\mathbb{Q}^+,\mathbb{R}^+$	the set of all positive elements of $\mathbb{Z},\mathbb{Q},\mathbb{R}$	Example 1.1.1
$\mathbb{Z}^-,\mathbb{Q}^-,\mathbb{R}^-$	the set of all negative elements of $\mathbb{Z},\mathbb{Q},\mathbb{R}$	Example 1.1.1
$\mathbb{Z}^,\mathbb{Q}^,\mathbb{R}^,\mathbb{C}^$	the set of all nonzero elements of $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$	Example 1.1.1
$\mathbb{M}_{m\times n}(S)$	the set of all $m \times n$ matrices over $S$	Definition 1.1.3
$\mathbb{M}_n(S)$	the set of all $n \times n$ matrices over $S$	Definition 1.1.3
$A\subseteq B$	$A$ is a subset of the $B$	Definition 1.1.4
$A\subsetneq B$	$A$ is a proper subset of $B$	Definition 1.1.4
$P(A)$	the power set of $A$	Definition 1.1.5
$A\cap B$	the intersection of $A$ and $B$	Definition 1.1.6
$A\cup B$	the union of $A$ and $B$	Definition 1.1.6
$A - B$	the difference of $A$ and $B$	Definition 1.1.6
$\bigcup_{i\in I}A_i$	$\{x: x\in A_i \text{ for some } i\in I\}$	Definition 1.1.6
$\bigcap_{i\in I}A_i$	$\{x: x\in A_i \text{ for every } i\in I\}$	Definition 1.1.6
$A\times B$	the direct product of $A$ and $B$	Definition 1.1.7
$f:S\to T$	function $f$ from $S$ to $T$	Definition 1.2.1
$f(U)$	the image of a set $U$ under $f$	Definition 1.2.1
$f^{\leftarrow}(V)$	the preimage of a set $V$ under $f$	Definition 1.2.1
$f\circ g$	the composition of $f$ with $g$	Definition 1.2.3
$1_S$	the identity function on $S$	Definition 1.2.3
$f^{-1}$	the inverse of $f$	Theorem 1.2.2
$\|S\|$	the cardinality of $S$	Definition 1.3.1
$\langle S, *\rangle$	binary structure	Definition 2.1.1
$e$	the identity element in a binary structure/group	Definition 2.1.4
$\det A$	the determinant of $A$	Definition 2.4.1
$GL(n,\mathbb{R})$	the general linear group of degree $n$ over $\mathbb{R}$	Definition 2.4.1
$I_n$	the $n\times n$ identity matrix	Theorem 2.4.1
$e_G$	the identity element in a group $G$	Convention 2.5.1
$a^{-1}$	the inverse of $a$ in a group	Convention 2.5.1
$-a$	the inverse of $a$ in an abelian group	Item
$n\mathbb{Z}$	$\{nm\,:\,m\in \mathbb{Z}\}$	Example 2.6.1
$a\equiv_n b$	$a$ is congruent to $b$ mod $n$	Definition 2.6.1
$R_n(a)$	the remainder when $a$ is divided by $n$	Definition 2.6.2
$+_n$	addition modulo $n$	Definition 2.6.3
$\mathbb{Z}_n$	the cyclic group of order $n$	Example 2.6.3
$\mathbb{Z}_n^{\times}$	$\{a\in \mathbb{Z}_n\,:\,\gcd(a,n)=1\}$	Definition 2.6.7
$F$	the set of all functions from $\mathbb{R}$ to $\mathbb{R}$	Example 2.6.6
$B$	the set of all bijections from $\mathbb{R}$ to $\mathbb{R}$	Example 2.6.7
$Z(G)$	the center of a group $G$	Exercise 2.8.9
$C^1$	the set of all differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ whose derivatives are continuous	Item 6
$C^0$	the set of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$	Item 7
$c_a$	conjugation by $a$	Example 3.2.2
$G\simeq G'$	$G$ is isomorphic to $G'$	Definition 3.3.1
$G\not \simeq G'$	$G$ is not isomorphic to $G'$	Definition 3.3.1
$H\leq G$	$H$ is a subgroup of $G$	Definition 4.1.1
$H\not \leq G$	$H$ is not a subgroup of $G$	Definition 4.1.1
$\langle a \rangle$	the (cyclic) subgroup generated by $a$	Definition 5.1.2
$o(a)$	the order of element $a$	Definition 5.1.2
$S_A$	the set of all permutations on $A$	Definition 6.1.3
$S_n$	the symmetric group on $n$ letters	Definition 6.2.1
$A_n$	the alternating group on $n$ letters	Definition 6.3.2
$\lambda_a$	left multiplication by $a$	Definition 6.4.1
$\rho_a$	right multiplication by $a$	Definition 6.4.1
$\mapsto$	maps to	Paragraph
$D_n$	the dihedral group of order $2n$	Definition 6.5.1
$xRy$	$x$ is related to $y$	Definition 7.1.2
$x\not R y$	$x$ is not related to $y$	Definition 7.1.2
$[x]$	the equivalence class of $x$	Definition 7.1.4
$a\sim_L b$	$a^{-1}b\in H\text{,}$ where $H\leq G$ is specified	Definition 7.2.1
$a\sim_R b$	$ab^{-1}\in H\text{,}$ where $H\leq G$ is specified	Definition 7.2.1
$aH, a+H$	the left coset of $H$ containing $a$	Definition 7.2.2
$Ha, H+a$	the right coset of $H$ containing $a$	Definition 7.2.2
$\Leftrightarrow$	if and only if	Note 7.2
$H\unlhd G$	$H$ is a normal subgorup of $G$	Definition 7.2.3
$G/H$	the set of all left cosets of $H$ in $G$	Definition 7.2.4
$(G:H)$	$\|G/H\|$	Definition 7.3.1
$aHb$	$\{ahb\,:h\in H\}$	Definition 8.2.1
$\text{Ker} \phi$	the kernel of $\phi$	Definition 8.2.3
$G/N$	the factor group $G/N\text{,}$ when $N\unlhd G$	Definition 8.3.1
$\Psi$	the canonical epimorphism from $G$ to $G/N$	Definition 8.3.3
$S^1$	the unit circle $\{e^{i\theta} \,:\, \theta\in f\}$ in the complex plane	Paragraph

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