Appendix B Index of Notation
| Symbol | Description | Location |
|---|---|---|
| \(\lgcnot A\) | logical negation of statement \(A\) | Item |
| \(A \lgcand B\) | logical conjunction of statements \(A\) and \(B\) | Item |
| \(A \lgcor B\) | logical disjunction of statements \(A\) and \(B\) | Item |
| \(A \lgccond B\) | logical conditional where statement \(A\) implies statement \(B\) | Item |
| \(A \lgcbicond B\) | logical biconditional where each of statements \(A\) and \(B\) implies the other | Item |
| \(A \lgcimplies B\) | statement \(A\) logically implies statement \(B\text{,}\) so that conditional \(A \lgccond B\) is a tautology | Item |
| \(A \lgcequiv B \) | statements \(A\) and \(B\) are equivalent | Item |
| \(\boolnot{x}\) | 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 |
| \(A_1, A_2, \dotsc, A_m \therefore C \) | an argument with premises \(A_1, A_2, \dotsc, A_m\) and conclusion \(C\) | Item |
| \(\begin{array}{c} A_1 \\ A_2 \\ \vdots \\ A_m \\ \hline C \end{array}\) | an argument with premises \(A_1, A_2, \dotsc, A_m\) and conclusion \(C\) | Item |
| \(x \in S\) | object \(x\) is an element of set \(S\) | Item |
| \(\{a,b,c,\dotsc\}\) | 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 |
| \(A \subsetneqq B\) | set \(A\) is a proper subset of set \(B\) | Item |
| \(\cmplmnt{A}\) | the complement of \(A\) relative to some universal set | Item |
| \(B \relcmplmnt A\) | the complement of \(A\) relative to some universal set | Item |
| \(\I\) | the set of irrational real numbers | Item |
| \(A \union B\) | the union of sets \(A\) and \(B\) | Item |
| \(A \intersection B\) | the intersection of \(A\) and \(B\) | Item |
| \(A \disjunion B\) | the disjoint union of sets \(A\) and \(B\) | Item |
| \(A \cartprod B\) | the Cartesian product of \(A\) and \(B\) | Item |
| \(A^n\) | the Cartesian product \(A \cartprod A \cartprod \dotsb \cartprod A\) involving \(n\) copies of \(A\) | Item |
| \(\words{\Sigma}\) | the set of words using alphabet set \(\Sigma\) | Item |
| \(\length{w}\) | length of the word \(w \in \words{\Sigma}\) | Item |
| \(\words{\Sigma}_n\) | for \(n \in \N\text{,}\) the subset of \(\words{\Sigma}\) consisting of all words of length \(n\) | Item |
| \(\emptyword\) | the empty word | Item |
| \(\powset{A}\) | the power set of the set \(A\) | Item |
| \(\funcdef{f}{A}{B}\) | \(f\) is a function with domain \(A\) and codomain \(B\) | Item |
| \(f(a) = b\) | function \(\funcdef{f}{A}{B}\) 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 |
| \(\funcgraph{f}\) | graph of function \(f\) | Item |
| \(f(A)\) | the image of function \(\funcdef{f}{A}{B}\) | Item |
| \(f(A')\) | the image of function \(\funcdef{f}{A}{B}\) on a subset \(A' \subseteq A\) | Item |
| \(\sfuncdef{f}{A}{B}\) | function \(f\) is surjective | Item |
| \(\ifuncdef{f}{A}{B}\) | function \(f\) is injective | Item |
| \(\funcdef{\id_A}{A}{A}\) | the identity function on on set \(A\) | Item |
| \(\funcdef{\inclfunc{A}{X}}{A}{X}\) | the inclusion function on subset \(A \subseteq X\) | Item |
| \(\funcdef{\projfunc{i}}{A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n}{A_i}\) | the projection function onto the \(\nth[i]\) factor \(A_i\) in the Cartesian product
\begin{equation*}
A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n
\end{equation*}
|
Paragraphs |
| \(\funcdef{\proj_i}{A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n}{A_i}\) | alternative notation for \(\projfunc{i}\) | Paragraphs |
| \(\funcres{f}{A}\) | restriction of function \(\funcdef{f}{X}{Y}\) to subset \(A \subseteq X\) | Item |
| \(\altfuncres{f}{A}\) | alternative domain restriction notation | Item |
| \(\res_A^X f\) | alternative domain restriction notation | Item |
| \(g \funccomp f\) | the composition of functions \(f\) and \(g\) | Item |
| \(\funcinvimg{f}{C}\) | the inverse image of the subset \(C \subseteq B\) under the function \(\funcdef{f}{A}{B}\) | Item |
| \(\funcdef{\inv{f}}{B}{A}\) | the inverse function associate to bijective function \(\funcdef{f}{A}{B}\) | Item |
| \(\natnumlt{m}\) | the set of natural numbers that are less than \(m\) | Item |
| \(a_k\) | \(\nth[k]\) 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 |
| \(\card{A}\) | cardinality of the set \(A\) | Item |
| \(\cardop A\) | alternative notation for the cardinality of the set \(A\) | Item |
| \(\ncardop\{\dots\}\) | alternative notation for the cardinality of the set defined by \(\{\dots\}\) | Item |
| \(\card{A} = \infty\) | set \(A\) is infinite | Item |
| \(\card{A} \lt \infty\) | set \(A\) is finite | Item |
| \(\deg v\) | degree of vertex \(v\) | Item |
| \(\card{E}\) | the number of edges in the graph \(G = (V,E)\) | Item |
| \(G' \subgraph G\) | graph \(G'\) is a subgraph of graph \(G\) | Item |
| \(K_n\) | the unique complete graph with \(n\) vertices | Item 1 |
| \(a \mathrel{R} b\) | element \(a \in A\) is related to element \(b \in B\) by relation \(R\) | Item |
| \(R_1 \union R_2\) | union of relations \(R_1,R_2\) | Item |
| \(R_1 \intersection R_2\) | intersection of relations \(R_1,R_2\) | Item |
| \(\cmplmnt{R}\) | complement of relation \(R\) | Item |
| \(a\ \not R\ b\) | alternative notation for \(a \mathrel{\cmplmnt{R}} b\) | Item |
| \(\inv{R}\) | inverse of the relation \(R\) | Item |
| \(a \mathrel{\emptyset} b\) | the empty relation between elements \(a\) and \(b\) (always false) | Item |
| \(a \mathrel{U} b\) | the universal relation between elements \(a\) and \(b\) (always true) | Item |
| \(a \equiv b\) | \(a\) is related to \(b\) by the equivalence relation \(\mathord{\equiv}\text{;}\) 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 |
| \(\eqclass{a}\) | the equivalence class of the element \(a \in A\) relative to some specific equivalence relation on \(A\) | Item |
| \(A / \mathord{\equiv} \) | the quotient of \(A\) relative to equivalence relation \(\mathord{\equiv}\) | Item |
| \(a \mathord{\partorder} b\) | \(a\) is related to \(b\) by the partial order \(\mathord{\partorder}\text{;}\) in other words, \(a\) is somehow “smaller than or same size as” \(b\) | Item |
| \(a \partorderstrict b\) | \(a \partorder b\) but \(a \neq b\) | Item |
| \(n!\) | factorial \(n! = n (n - 1) (n - 2) \dotsm 2 \cdot 1\) | Item |
| \(\permutation{n}{k}\) | the number of permutations of size \(k\) taken from a set of size \(n\) | Item |
| \(\permutationalt{n}{k}\) | alternative notation for \(\permutation{n}{k}\) | Item |
| \(\permutationaltalt{n}{k}\) | alternative notation for \(\permutation{n}{k}\) | Item |
| \(\combination{n}{k}\) | the number of combination of size \(k\) taken from a set of size \(n\) | Item |
| \(\combinationalt{n}{k}\) | alternative notation for \(\combination{n}{k}\) | Item |
| \(\combinationaltalt{n}{k}\) | alternative notation for \(\combination{n}{k}\) | Item |
| \(\binom{n}{k}\) | the \(\nth[k]\) coefficient in the expansion of \((x + y)^n\) | Item |
| \(\binom{n}{i_1,i_2,\dotsc,i_m}\) | the coefficient on the term \(x_1^{i_1} x_2^{i_2} \dotsm x_m^{i_m}\) in the expansion of \((x_1 + x_2 + \dotsm + x_m)^n\) | Item |
