ads Notation

Appendix E Notation

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
$\| A \|$	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
${}$	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 1.2.4
$B - A$	The complement of $A$ relative to $B$	Definition 1.2.10
$A^{c}$	The complement of set $A$ relative to the universe.	Definition 1.2.10
$A \oplus B$	The symmetric difference of $A$ and $B .$	Definition 1.2.15
$A \times B$	The cartesian product of $A$ with $B .$	Definition 1.3.1
$P (A)$	The power set of $A,$ the set of all subsets of $A .$	Definition 1.3.3
$n!$	$n$ factorial, the product of the first $n$ positive integers	Definition 2.2.5
$(\binom{n}{k})$	$n$ choose $k,$ the number of $k$ element subsets of an $n$ element set.	Definition 2.4.3
$p \land q$	the conjunction, $p and q$	Definition 3.1.3
$p \lor q$	the disjunction, $p or q$	Definition 3.1.4
$\neg p$	the negation of $p,$ “not $p$ ”	Definition 3.1.5
$p \to q$	The conditional proposition If $p$ then $q .$	Definition 3.1.6
$p \leftrightarrow q$	The biconditional proposition $p$ if and only if $q$	Definition 3.1.12
$1$	symbol for a tautology	Definition 3.3.2
$0$	symbol for a contradiction	Definition 3.3.4
$r ⟺ s$	$r$ is logically equivalent to $s$	Definition 3.3.6
$r \Rightarrow s$	$r$ implies $s$	Definition 3.3.11
$p ∣ q$	the Sheffer Stroke of $p$ and $q$	Definition 3.3.14
$◼$	Symbol that denotes the end of a proof. Can be replaced with QED	Paragraph
$T_{p}$	the truth set of $p$	Definition 3.6.3
$(\exists n)_{U} (p (n))$	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 3.8.3
$0 0_{m \times n}$	the $m$ by $n$ zero matrix	Item
$I_{n}$	The $n \times n$ identity matrix	Definition 5.2.4
$A^{- 1}$	$A$ inverse, the multiplicative inverse of $A$	Definition 5.2.5
$det A or \| A \|$	The determinant of $A,$ 2 by 2 case	Definition 5.2.7
$a ∣ b$	$a$ divides $b,$ or $a$ divides evenly into $b$	Definition 6.1.5
$x s y$	$x$ is related to $y$ through the relation $s$	Paragraph
$r s$	the composition of relation $r$ with relation $s$	Definition 6.1.9
$[a]$	The equivalence class of a	Definition 6.3.13
$A / r$	Partition of $A$ with respect to an equivalence relation $r$	Definition 6.3.13
$a \equiv_{n} b$	$a$ is congruent to $b$ modulo $n$	Definition 6.3.15
$a \equiv b (mod n)$	$a$ is congruent to $b$ modulo $n$	Definition 6.3.15
$r^{+}$	The transitive closure of $r$	Definition 6.5.1
$f : A \to B$	A function, $f,$ from $A$ into $B$	Definition 7.1.1
$B^{A}$	The set of all functions from $A$ into $B$	Definition 7.1.5
$f (a)$	The image of $a$ under $f$	Definition 7.1.7
$f (X)$	Range of function $f : X \to Y$	Definition 7.1.8
$χ_{S}$	Characteristic function of the set $S$	Exercise 7.1.5.4
$\| A \| = n$	$A$ has cardinality $n$	Definition 7.2.7
$(g \circ f) (x) = g (f (x))$	The composition of $g$ with $f$	Definition 7.3.2
$f \circ f = f^{2}$	the “square” of a function.	Definition 7.3.5
$i or i_{A}$	The identity function (on a set $A$ )	Definition 7.3.8
$f^{- 1}$	The inverse of function $f$ read “ $f$ inverse”	Definition 7.3.11
$l o g_{b} a$	Logarithm, base $b$ of $a$	Definition 8.4.4
		Definition 8.5.1
$S ↑$	$S$ pop	Definition 8.5.3
$S ↓$	$S$ push	Definition 8.5.3
$S * T$	🔗 Convolution of sequences $S$ and $T$	Definition 8.5.3
$S ↑ p$	🔗 Multiple pop operation on $S$	Definition 8.5.6
$S ↓ p$	🔗 Multiple push operation on $S$	Definition 8.5.6
$K_{n}$	A complete undirected graph with $n$ vertices.	Definition 9.1.10
$d e g (v), i n d e g (v), o u t d e g (v)$	degree, indegree and outdegree of vertex $v$	Definition 9.1.30
$e (v)$	The eccentricity of a vertex	Paragraph
$d (G)$	The diameter of graph $G$	Paragraph
$r (G)$	The radius of graph $G$	Paragraph
$C (G)$	The center of graph $G$	Paragraph
$Q_{n}$	🔗 the $n$ -cube	Definition 9.4.17
$V (f)$	The value of flow $f$	Definition 9.5.21
$P_{n}$	a path graph of length $n$	Definition 9.6.4
$χ (G)$	the chromatic number of $G$	Definition 9.6.15
$C_{n}$	A cycle with $n$ edges.	Definition 10.1.1
$*$	generic symbol for a binary operation	Definition 11.1.1
$s t r i n g 1 + s t r i n g 2$	The concatenation of $s t r i n g 1$ and $s t r i n g 2$	Item a
$[G; *]$	a group with elements $G$ and binary operation $*$	Definition 11.2.3
$gcd (a, b)$	the greatest common divisor of $a$ and $b$	Definition 11.4.4
$Z_{n}$	the integers modulo $n$	Definition 11.4.12
$a +_{n} b$	the mod $n$ sum of $a$ and $b$	Definition 11.4.13
$a \times_{n} b$	the mod $n$ product of $a$ and $b$	Definition 11.4.14
$Z_{n}$	The Additive Group of Integer Modulo $n$	Definition 11.4.18
$U_{n}$	The Multiplicative Group of Integer Modulo $n$	Definition 11.4.19
$W \leq V$	$W$ is a subsystem of $V$	Definition 11.5.1
$⟨ a ⟩$	the cyclic subgroup generated by $a$	Definition 11.5.6
$o r d (a)$	Order of a	Definition 11.5.9
$V_{1} \times V_{2} \times \dots \times V_{n}$	The direct product of algebraic structures $V_{1}, V_{2}, \dots, V_{n}$	Definition 11.6.1
$G_{1} \times G_{2}$	The direct product of groups $G_{1}$ and $G_{2}$	Definition 11.6.3
$G_{1} ≅ G_{2}$	$G_{1}$ is isomorphic to $G_{2}$	Definition 11.7.9
		Definition 12.3.6
$d i m (V)$	The dimension of vector space $V$	Definition 12.3.18
$0 0$	least element in a poset	Definition 13.1.7
$1 1$	greatest element in a poset	Definition 13.1.7
$D_{n}$	the set of divisors of integer $n$	Definition 13.1.9
$a \lor b$	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 13.3.6
$[B; \lor, \land, \bar{}]$	a boolean algebra with operations join, meet and complementation	Definition 13.3.8
		Definition 13.3.12
$M_{δ_{1} δ_{2} \dots δ_{k}}$	the minterm generated by $x_{1}, x_{2}, \dots, x_{k},$ where $y_{i} = x_{i}$ if $δ_{i} = 1$ and $y_{i} = \bar{x_{i}}$ if $δ_{i} = 0$	Definition 13.6.3
$A^{*}$	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
$λ$	The empty string	Definition 14.2.1
$s_{1} + s_{2}$	The concatenation of strings $s_{1}$ and $s_{2}$	Definition 14.2.4
$L (G)$	Language created by phrase structure grammar $G$	Definition 14.2.15
$(S, X, Z, w, t)$	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
		Definition 15.1.1
$a * H, H * a$	the left and right cosets generated by $a$	Definition 15.2.2
$G / H$	The factor group G mod H.	Definition 15.2.20
$S_{A}$	The group of permutations of the set $A$	Definition 15.3.5
$S_{n}$	The group of permutations on a set with $n$ elements	Definition 15.3.5
$A_{n}$	The Alternating Group	Definition 15.3.18
$D_{n}$	The $n$ th dihedral group	Definition 15.3.26
$H ◃ G$	$H$ is a normal subgroup of $G$	Definition 15.4.5
$k e r θ$	the kernel of homomorphism $θ$	Definition 15.4.19
$d_{H} (a, b)$	Hamming distance between $a$ and $b$	Definition 15.5.3
$[R; +, \cdot]$	a ring with domain $R$ and operations $+$ and $\cdot$	Definition 16.1.1
$U (R)$	the set of units of a ring $R$	Definition 16.1.10
		Definition 16.1.13
		Definition 16.1.20
$D$	a generic integral domain	Definition 16.1.23
$deg f (x)$	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
$R [[x]]$	the set of all powers series in $R$	Definition 16.5.1
$\overset{`}{x}, \overset{´}{x}$	pre and post values of a variable $x$	Definition A.2.1
$M (A)_{i, j}$	The $i, j$ minor of $A$	Definition C.1.2
$C (A)_{i, j}$	The $i, j$ cofactor of $A$	Definition C.1.4
$det (A) o r \| A \|$	The determinant of $A$	Definition C.1.6

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