17.4: D - Notation
- Page ID
- 80583
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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 |
|---|---|---|
| \(x\in A\) | \(x\) is an element of \(A\) | Paragraph |
| \(x\notin A\) | \(x\) is not an element of \(A\) | Paragraph |
| \(\mathbb{P}\) | the positive integers | Item |
| \(\mathbb{N}\) | the natural numbers | Item |
| \(\mathbb{Z}\) | the integers | Item |
| \(\mathbb{Q}\) | the rational numbers | Item |
| \(\mathbb{R}\) | the real numbers | Item |
| \(\mathbb{C}\) | the complex numbers | Item |
| \(|A|\) | The number of elements in a finite set \(A\). | Definition 1.1.2 |
| \(A\subseteq B\) | \(A\) is a subset of \(B\). | Definition 1.1.3 |
| \(\emptyset\) | the empty set | Paragraph |
| \(A \cap B\) | The intersection of \(A\) and \(B\). | Definition 1.2.1 |
| \(A \cup B\) | The union of \(A\) and \(B\). | Definition 1.2.3 |
| \(B-A\) | The complement of \(A\) relative to \(B\) | Definition 1.2.5 |
| \(A^c\) | The complement of set \(A\) relative to the universe. | Definition 1.2.5 |
| \(A\oplus B\) | The symmetric difference of \(A\) and \(B\). | Definition 1.2.6 |
| \(A\times B\) | The cartesian product of \(A\) with \(B\). | Definition 1.3.1 |
| \(\mathcal{P}(A)\) | The power set of \(A\), the set of all subsets of \(A\). | Definition 1.3.2 |
| \(n!\) | \(n\) factorial, the product of the first \(n\) positive integers | Definition 2.2.1 |
| \(\binom{n}{k}\) | \(n\) choose \(k\), the number of \(k\) element subsets of an \(n\) element set. | Definition 2.4.1 |
| \(p \land q\) | the conjunction, \(p\) and \(q\) | Definition 3.1.2 |
| \(p \lor q\) | the disjunction, \(p\) or \(q\) | Definition 3.1.3 |
| \(\neg p\) | the negation of \(p\), "not \(p\)" | Definition 3.1.4 |
| \(p \to q\) | The conditional proposition If \(p\) then \(q\). | Definition 3.1.5 |
| \(p \leftrightarrow q\) | The biconditional proposition \(p\) if and only if \(q\) | Definition 3.1.8 |
| \(1\) | symbol for a tautology | Definition 3.3.1 |
| \(0\) | symbol for a contradiction | Definition 3.3.2 |
| \(r \iff s\) | \(r\) is logically equivalent to \(s\) | Definition 3.3.3 |
| \(r \Rightarrow s\) | \(r\) implies \(s\) | Definition 3.3.4 |
| \(p|q\) | the Sheffer Stroke of \(p\) and \(q\) | Definition 3.3.5 |
| \(T_p\) | the truth set of \(p\) | Definition 3.6.2 |
| \((\exists n)_U(p(n))\) | The statement that \(p(n)\) is true for at least one value of \(n\) | Definition 3.8.1 |
| \((\forall n)_U(p(n))\) | The statement that \(p(n)\) is always true. | Definition 3.8.2 |
| \(\pmb{0}_{m\times n}\) | the \(m\) by \(n\) zero matrix | Item |
| \(I_n\) | The \(n\times n\) identity matrix | Definition 5.2.2 |
| \(A^{-1}\) | \(A\) inverse, the multiplicative inverse of \(A\) | Definition 5.2.3 |
| \(\det\: A\) or \(|A|\) | The determinant of \(A\), 2 by 2 case | Definition 5.2.4 |
| \(a|b\) | \(a\) divides \(b\), or \(a\) divides evenly into \(b\) | Definition 6.1.3 |
| \(xsy\) | \(x\) is related to \(y\) through the relation \(s\) | Paragraph |
| \(rs\) | the composition of relation \(r\) with relation \(s\) | Definition 6.1.4 |
| \(a \equiv_n b\) | \(a\) is congruent to \(b\) modulo \(n\) | Definition 6.3.7 |
| \(a \equiv b (\textrm{mod } n)\) | \(a\) is congruent to \(b\) modulo \(n\) | Definition 6.3.7 |
| \(c(a)\) | the equivalence class of \(a\) under \(r\) | Item |
| \(r^+\) | The transitive closure of \(r\) | Definition 6.5.1 |
| \(f:A \rightarrow B\) | A function, \(f\), from \(A\) into \(B\) | Definition 7.1.1 |
| \(B^A\) | The set of all functions from \(A\) into \(B\) | Definition 7.1.2 |
| \(f(a)\) | The image of \(a\) under \(f\) | Definition 7.1.3 |
| \(f(X)\) | Range of function \(f:X \rightarrow Y\) | Definition 7.1.4 |
| \(\mathcal{X}s\) | Characteristic function of the set \(S\) | Exercise 7.1.4 |
| \(|A|=n\) | \(A\) has cardinality \(n\) | Definition 7.2.4 |
| \((g \circ f)(x) = g(f(x))\) | The composition of \(g\) with \(f\) | Definition 7.3.2 |
| \(f \circ f = f^2\) | the "square" of a function. | Definition 7.3.3 |
| \(i\) or \(i_A\) | The identity function (on a set \(A\)) | Definition 7.3.4 |
| \(f^{-1}\) | The inverse of function \(f\) read "\(f\) inverse" | Definition 7.3.5 |
| \(log_b a\) | Logarithm, base \(b\) of \(a\) | Definition 8.4.2 |
| \(S\uparrow\) | \(S\) pop | Definition 8.5.2 |
| \(S\downarrow\) | \(S\) push | Definition 8.5.2 |
| \(S*T\) | Convolution of sequences \(S\) and \(T\) | Definition 8.5.2 |
| \(S\uparrow p\) | Multiple pop operation on \(S\) | Definition 8.5.3 |
| \(S\downarrow p\) | Multiple push operation on \(S\) | Definition 8.5.3 |
| \(K_n\) | A complete undirected graph with \(n\) vertices | Definition 9.1.4 |
| \(deg(v), indeg(v), outdeg(v)\) | degree, indegree, and outdegree of vertex \(v\) | Definition 9.1.9 |
| \(e(v)\) | The eccentricity of a vertex | Definition 9.3.1 |
| \(d(G)\) | The diameter of graph \(G\) | Definition 9.3.2 |
| \(r(G)\) | The radius of graph \(G\) | Definition 9.3.3 |
| \(C(G)\) | The center of graph \(G\) | Definition 9.3.4 |
| \(Q_n\) | the \(n\)-cube | Definition 9.4.3 |
| \(V(f)\) | The value of flow \(f\) | Definition 9.5.3 |
| \(P_n\) | a path graph of length \(n\) | Definition 9.6.2 |
| \(\chi(G)\) | the chromatic number of \(G\) | Definition 9.6.3 |
| \(C_n\) | A cycle with \(n\) edges. | Definition 10.1.1 |
| \(*\) | generic symbol for a binary operation | Definition 11.1.1 |
| \(string1 + string2\) | The concatenation of \(string1\) and \(string2\) | Item |
| \([G;*]\) | a group with elements \(G\) and binary operation \(*\) | Definition 11.2.1 |
| \(\gcd(a,b)\) | the greatest common divisor of \(a\) and \(b\) | Definition 11.4.1 |
| \(a +_n b\) | the mod \(n\) sum of \(a\) and \(b\) | Definition 11.4.3 |
| \(a \times_n b\) | the mod \(n\) product of \(a\) and \(b\) | Definition 11.4.4 |
| \(\mathbb{Z}_n\) | The Additive Group of Integer Modulo \(n\) | Definition 11.4.5 |
| \(\mathbb{U}_n\) | The Multiplicative Group of Integer Modulo \(n\) | Definition 11.4.6 |
| \(W\leq V\) | \(W\) is a subsystem of \(V\) | Definition 11.5.1 |
| \(\langle a \rangle\) | the cyclic subgroup generated by \(a\) | Definition 11.5.2 |
| \(ord(a)\) | Order of a | Definition 11.5.3 |
| \(V_1\times V_2 \times \cdots \times V_n\) | The direct product of algebraic structures \(V_1, V_2, \dots , V_n\) | Definition 11.6.1 |
| \(G_1 \cong G_2\) | \(G_1\) is isomorphic to \(G_2\) | Definition 11.7.2 |
| \(dim(V)\) | The dimension of vector space \(V\) | Definition 12.3.6 |
| \(\pmb{0}\) | least element in a poset | Definition 13.1.5 |
| \(\pmb{1}\) | greatest element in a poset | Definition 13.1.5 |
| \(D_n\) | the set of divisors of integer \(n\) | Definition 13.1.6 |
| \(a \lor b\) | the join, or least upper bound of \(a\) and \(b\) | Definition 13.2.1 |
| \(a \land b\) | the meet, or greatest lower bound of \(a\) and \(b\) | Definition 13.2.1 |
| \([L;\lor,\land]\) | A lattice with domain having meet and join operations | Definition 13.2.2 |
| \(\bar{a}\) | The complement of lattice element \(a\) | Definition 13.3.4 |
| \([B; \lor , \land, \bar{\hspace{5 mm}}]\) | a boolean algebra with operations join, meet and complementation | Definition 13.3.5 |
| \(M_{\delta_1 \delta_2 \cdots \delta_k}\) | the minterm generated by \(x_1, x_2, \ldots , x_k\text{,}\) where \(y_i=x_i\) if \(\delta_i = 1\) and \(y_i=\bar{x_i}\) if \(\delta_i = 0\) | Definition 13.6.2 |
| \(A^*\) | The set of all strings over an alphabet \(A\) | Definition 14.2.1 |
| \(A^n\) | The set of all strings of length \(n\) over an alphabet \(A\) | Definition 14.2.1 |
| \(\lambda\) | The empty string | Definition 14.2.1 |
| \(s_1+s_2\) | The concatenation of strings \(s_1\) and \(s_2\) | Definition 14.2.2 |
| \(L(G)\) | Language created by phrase structure grammar \(G\) | Definition 14.2.5 |
| \((S, X, Z, w, t)\) | A finite-state machine with states \(S\), input alphabet \(X\), output alphabet \(X\), and output function \(w\) and next-state function \(t\) | Definition 14.3.1 |
| \(m(M)\) | The machine of monoid \(M\) | Definition 14.5.1 |
| \(a*H, H*a\) | the left and right cosets generated by \(a\) | Definition 15.2.1 |
| \(G/H\) | The factor group \(G\) mod \(H\). | Definition 15.2.4 |
| \(S_A\) | The group of permutations of the set \(A\) | Definition 15.3.1 |
| \(S_n\) | The group of permutations on a set with \(n\) elements | Definition 15.3.1 |
| \(A_n\) | The Alternating Group | Definition 15.3.3 |
| \(\mathcal{D}_n\) | The \(n\)th dihedral group | Definition 15.3.4 |
| \(H \triangleleft G\) | \(H\) is a normal subgroup of \(G\) | Definition 15.4.1 |
| \(ker\theta\) | the kernel of homomorphism \(\theta\) | Definition 15.4.4 |
| \([R; +, \cdot]\) | a ring with domain \(R\) and operations \(+\) and \(\cdot\). | Definition 16.1.1 |
| \(U(R)\) | the set of units of a ring \(R\) | Definition 16.1.4 |
| \(D\) | a generic integral domain | Definition 16.1.7 |
| \(\textrm{deg }f(x)\) | the degree of polynomial \(f(x)\) | Definition 16.3.1 |
| \(R[x]\) | the set of all polynomials in \(x\) over \(R\) | Definition 16.3.1 |
| \(R\left[\left[x\right]\right]\) | the set of all power series in \(R\) | Definition 16.5.1 |
| \(\grave x, \acute x\) | pre and post values of a variable \(x\) | Definition 17.1.2.1 |
| \(M(A)_{i,j}\) | The \(i\), \(j\) minor of \(A\) | Definition 17.3.1.2 |
| \(C(A)_{i,j}\) | The \(i\), \(j\) cofactor of \(A\) | Definition 17.3.1.3 |
| \(\det(A)\) or \(\lvert A \rvert\) | The determinant of \(A\) | Definition 17.3.1.4 |