EF Index of Notation

Appendix B Index of Notation

Symbol	Description	Location
$\neg A$	logical negation of statement $A$	Item
$A \land B$	logical conjunction of statements $A$ and $B$	Item
$A \lor B$	logical disjunction of statements $A$ and $B$	Item
$A \to B$	logical conditional where statement $A$ implies statement $B$	Item
$A \leftrightarrow B$	logical biconditional where each of statements $A$ and $B$ implies the other	Item
$A \Rightarrow B$	statement $A$ logically implies statement $B,$ so that conditional $A \to B$ is a tautology	Item
$A \Leftrightarrow B$	statements $A$ and $B$ are equivalent	Item
$x^{'}$	Boolean negation	Item
$A (x)$	a predicate statement $A$ whose truth value depends on the free variable $x$	Item
$A (x, y)$	a predicate statement $A$ whose truth value depends on the free variables $x$ and $y$	Item
$\forall x$	the universal quantifier applied to the free variable $x$	Item
$\exists x$	the existential quantifier applied to the free variable $x$	Item
$A_{1}, A_{2}, \dots, A_{m} ∴ C$	an argument with premises $A_{1}, A_{2}, \dots, A_{m}$ and conclusion $C$	Item
$\begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \\ C \end{matrix}$	an argument with premises $A_{1}, A_{2}, \dots, A_{m}$ and conclusion $C$	Item
$x \in S$	object $x$ is an element of set $S$	Item
${a, b, c, \dots}$	a set defined by listing its elements, enclosed in braces	Paragraph
$N$	the set of natural numbers	Item
$Z$	the set of integers	Item
$Q$	the set of rational numbers	Item
$R$	the set of real numbers	Item
$\emptyset$	the empty set	Item
$A \subseteq B$	set $A$ is a subset of set $B$	Item
$A ⫋ B$	set $A$ is a proper subset of set $B$	Item
$A^{c}$	the complement of $A$ relative to some universal set	Item
$B ∖ A$	the complement of $A$ relative to some universal set	Item
$I$	the set of irrational real numbers	Item
$A \cup B$	the union of sets $A$ and $B$	Item
$A \cap B$	the intersection of $A$ and $B$	Item
$A ⊔ B$	the disjoint union of sets $A$ and $B$	Item
$A \times B$	the Cartesian product of $A$ and $B$	Item
$A^{n}$	the Cartesian product $A \times A \times \dots \times A$ involving $n$ copies of $A$	Item
$Σ^{*}$	the set of words using alphabet set $Σ$	Item
$\| w \|$	length of the word $w \in Σ^{*}$	Item
$Σ_{n}^{*}$	for $n \in N,$ the subset of $Σ^{*}$ consisting of all words of length $n$	Item
$\emptyset$	the empty word	Item
$P (A)$	the power set of the set $A$	Item
$f : A \to B$	$f$ is a function with domain $A$ and codomain $B$	Item
$f (a) = b$	function $f : A \to B$ associates the codomain element $b \in B$ to the domain element $a \in A$	Item
$a \mapsto b$	alternative notation for $f (a) = b$	Item
$Δ (f)$	graph of function $f$	Item
$f (A)$	the image of function $f : A \to B$	Item
$f (A^{'})$	the image of function $f : A \to B$ on a subset $A^{'} \subseteq A$	Item
$f : A ↠ B$	function $f$ is surjective	Item
$f : A ↪ B$	function $f$ is injective	Item
${id}_{A} : A \to A$	the identity function on on set $A$	Item
$ι_{A}^{X} : A \to X$	the inclusion function on subset $A \subseteq X$	Item
$ρ_{i} : A_{1} \times A_{2} \times \dots \times A_{n} \to A_{i}$	the projection function onto the $i^{th}$ factor $A_{i}$ in the Cartesian product $A_{1} \times A_{2} \times \dots \times A_{n}$	Paragraphs
${proj}_{i} : A_{1} \times A_{2} \times \dots \times A_{n} \to A_{i}$	alternative notation for $ρ_{i}$	Paragraphs
${f \|}_{A}$	restriction of function $f : X \to Y$ to subset $A \subseteq X$	Item
$f \| A$	alternative domain restriction notation	Item
${res}_{A}^{X} f$	alternative domain restriction notation	Item
$g \circ f$	the composition of functions $f$ and $g$	Item
$f^{- 1} (C)$	the inverse image of the subset $C \subseteq B$ under the function $f : A \to B$	Item
$f^{- 1} : B \to A$	the inverse function associate to bijective function $f : A \to B$	Item
$N_{< m}$	the set of natural numbers that are less than $m$	Item
$a_{k}$	$k^{th}$ term in a sequence	Item
${a_{k}}$	the collection of terms in a sequence	Item
${a_{k}}_{0}^{m}$	the collection of terms in a finite sequence	Item
${a_{k}}_{0}^{\infty}$	the collection of terms in an infinite sequence	Item
$\| A \|$	cardinality of the set $A$	Item
$card A$	alternative notation for the cardinality of the set $A$	Item
$# {\dots}$	alternative notation for the cardinality of the set defined by ${\dots}$	Item
$\| A \| = \infty$	set $A$ is infinite	Item
$\| A \| < \infty$	set $A$ is finite	Item
$\deg v$	degree of vertex $v$	Item
$\| E \|$	the number of edges in the graph $G = (V, E)$	Item
$G^{'} ⪯ G$	graph $G^{'}$ is a subgraph of graph $G$	Item
$K_{n}$	the unique complete graph with $n$ vertices	Item 1
$a R b$	element $a \in A$ is related to element $b \in B$ by relation $R$	Item
$R_{1} \cup R_{2}$	union of relations $R_{1}, R_{2}$	Item
$R_{1} \cap R_{2}$	intersection of relations $R_{1}, R_{2}$	Item
$R^{c}$	complement of relation $R$	Item
$a R̸ b$	alternative notation for $a R^{c} b$	Item
$R^{- 1}$	inverse of the relation $R$	Item
$a \emptyset b$	the empty relation between elements $a$ and $b$ (always false)	Item
$a U b$	the universal relation between elements $a$ and $b$ (always true)	Item
$a \equiv b$	$a$ is related to $b$ by the equivalence relation $\equiv;$ in other words, $a$ is somehow equivalent to $b$	Item
$m_{1} \equiv_{n} m_{2}$	integers $m_{1}, m_{2}$ are equivalent modulo $n$	Item
$[a]$	the equivalence class of the element $a \in A$ relative to some specific equivalence relation on $A$	Item
$A / \equiv$	the quotient of $A$ relative to equivalence relation $\equiv$	Item
$a ⪯ b$	$a$ is related to $b$ by the partial order $⪯;$ in other words, $a$ is somehow “smaller than or same size as” $b$	Item
$a ≺ b$	$a ⪯ b$ but $a \neq b$	Item
$n!$	factorial $n! = n (n - 1) (n - 2) \dots 2 \cdot 1$	Item
$P (n, k)$	the number of permutations of size $k$ taken from a set of size $n$	Item
$P_{k}^{n}$	alternative notation for $P (n, k)$	Item
${_{n} P}_{k}$	alternative notation for $P (n, k)$	Item
$C (n, k)$	the number of combination of size $k$ taken from a set of size $n$	Item
$C_{k}^{n}$	alternative notation for $C (n, k)$	Item
${_{n} C}_{k}$	alternative notation for $C (n, k)$	Item
$(\binom{n}{k})$	the $k^{th}$ coefficient in the expansion of $(x + y)^{n}$	Item
$(\binom{n}{i_{1}, i_{2}, \dots, i_{m}})$	the coefficient on the term $x_{1}^{i_{1}} x_{2}^{i_{2}} \dots x_{m}^{i_{m}}$ in the expansion of $(x_{1} + x_{2} + \dots + x_{m})^{n}$	Item

Elementary Foundations: An introduction to topics in discrete mathematics

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Appendix B Index of Notation