| 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 \(A_1 \cartprod \dotsb \cartprod A_n \)
|
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 |
| \(a \nmathrel{R} b \) |
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 \({(x_1 + x_2 + \dotsm + x_m)}^n \)
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Item |
