Notation
- Page ID
- 89327
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 |