Notation Glossary

Set Theory

$\forall$

universal quantifier

$\exists$

existential quantifier

$\neg$

logical negation (not)

$\land$

logical conjunction (and)

$\lor$

logical disjunction (or)

$\rimp$

logical implication (if … then …)

$\lrimp$

logical equivalence (if and only if)

$A = B$

set $A$ is equal to set $B$

$A \in B$

set $A$ is an element of set $B$

$\ZF$

Zermelo-Fraenkel Set Theory

$\AC$

Axiom of Choice

$\ZFC$

$\ZF + \AC$

$\{x : \varphi(x, p)\}$

Class-Builder

$\V$

the universal class

$\empt$

the empty set

$\{x \in X : \varphi(x, p)\}$

Set-Builder

$A \subseteq B$

set $A$ is a subset of set $B$

$A \subset B$

set $A$ is a proper subset of set $B$

$\{a, b\}$

the pair of $a$ and $b$

$\bigcup X$

the union of set $X$

$A \cup B$

the union of sets $A$ and $B$

$\{a\}$

the singleton of $a$

$\{a_1, \cdots, a_n\}$

Roster Notation

$\bigcap X$

the intersection of set $X$

$A \cap B$

the intersection of sets $A$ and $B$

$\bigsqcup X$

the disjoint union of set $X$

$A \sqcup B$

the disjoint union of sets $A$ and $B$

$A \setdif B$

the set difference between sets $A$ and $B$

$A \symdif B$

the symmetric difference between sets $A$ and $B$

$\powerset(X)$

the power set of set $X$

$(a_1, \cdots, a_n)$

the tuple of $a_1$, $\cdots$, $a_n$

$X_1 \times \cdots \times X_n$

the Cartesian product of $X_1$, $\cdots$, $X_n$

$X^n$

$\underbrace{X \times \cdots \times X}_{\text{$n$ times}}$

$R(x_1, \cdots, x_n)$

relation $R$ holds for $x_1$, $\cdots$, $x_n$

$\rel(X_1, \cdots, X_n)$

the set of all relations on $X_1$, $\cdots$, $X_n$

$x \,R\, y$

binary relation $R$ holds for $x$ and $y$

$\dom R$

the domain of binary relation $R$

$\im R$

the image of binary relation $R$

$\field R$

the field of binary relation $R$

$\id_X$

the identity on $X$

$[a]_\sim$ or $[a]$

the equivalence class of $a$ by equivalence relation $\sim$

$X / {\sim}$

the quotient set of $X$ by equivalence relation $\sim$

$R^{-1}$

the converse of binary relation $R$

$R \circ S$

the composition of binary relations $R$ and $S$

$R \restriction_A$

the left-restriction of binary relation $R$ to set $A$

$R \restriction^B$

the right-restriction of binary relation $R$ to set $B$

$R[A]$

the image of $A$ under binary relation $R$

$f(x)$

the value of function $f$ at $x$

$f : x \mapsto y$

function $f$ maps $x$ to $y$

$f : X \to Y$

function $f$ is from set $X$ to set $Y$

$\fun(X, Y)$

the set of all functions from set $X$ to set $Y$

$x * y$

the value of binary operation $*$ at $(x, y)$

$\inj(X, Y)$

the set of all injections from set $X$ to set $Y$

$\sur(X, Y)$

the set of all surjections from set $X$ to set $Y$

$\hom(X, Y)$

the set of all homomorphisms from set $X$ to set $Y$

$\mon(X, Y)$

the set of all monomorphisms from set $X$ to set $Y$

$\epi(X, Y)$

the set of all epimorphisms from set $X$ to set $Y$

$\iso(X, Y)$

the set of all isomorphisms from set $X$ to set $Y$

$\endo X$

the set of all endomorphisms on set $X$

$\aut X$

the set of all automorphisms on set $X$

$\leq$

a preorder

$<$

a strict preorder

$\max X$

the maximum of partially ordered set $X$

$\min X$

the minimum of partially ordered set $X$

$\upper A$

the set of all upper bounds of set $A$

$\lower A$

the set of all lower bounds of set $A$

$\sup A$

the supremum of set $A$

$\inf A$

the infimum of set $A$

$\init_u W$

the initial segment of well-ordered set $W$ by $u$

$\Ord$

the class of all ordinals

$\alpha < \beta$

ordinal $\alpha$ is smaller than ordinal $\beta$

$\alpha + 1$

the successor of ordinal $\alpha$

$\N$, $\omega$

the set of natural numbers

$0$, $1$, $2$, $\cdots$

natural numbers

$\langle s_\xi : \xi < \alpha \rangle$, $\langle s_\xi \rangle_{\xi < \alpha}$

a transfinite sequence

$\langle s_n : n \in \N\rangle$, $\langle s_n \rangle_{n \in \N}$

a countably infinite sequence

$s^\frown x$

the extension of transfinite sequence $s$ by $x$

$X \equ Y$

set $X$ is equinumerous to set $Y$

$X \lequ Y$

set $X$ is not greater than set $Y$

$X \lnequ Y$

set $X$ is smaller than set $Y$

$\Card$

the class of all cardinal numbers

$\alpha^+$

the cardinal successor of ordinal $\alpha$

$\kappa + \lambda$

the sum of cardinals $\kappa$ and $\lambda$

$\kappa \cdot \lambda$

the product of cardinals $\kappa$ and $\lambda$

$\kappa^\lambda$

the exponentiation of cardinal $\kappa$ by cardinal $\lambda$

$\powerset_\kappa(X)$

the set of all $\kappa$-sized subsets of set $X$

Abstract Algebra

$(X, *)$

a structure where set $X$ is equipped with binary opreation $*$ (magma, monoid, group, etc.)

$X \leq Y$

$X$ is substructure of $Y$ (submagma, submonoid, subgroup, etc.)

$\lvert M \rvert$

the order of magma $M$

$mA$, $Am$

the left and right coset of submagma $A$ by $m$

$M / A$, $M \backslash A$

the left and right coset quotient of magma $M$ by submagma $A$

$a^{-1}$

the inverse of $a$

$\inv M$

the invertible subset of magma $M$

$\ker f$

the kernel of group homomorphism $f$

$G \times H$

the direct product of groups $G$ and $H$

$\conj_g a$

the conjugation of $a$ by $g$ in a group

$\stackrel{\conj}{\sim}$

$a$ and $b$ are conjugate in a group

$G \unlhd H$

$H$ is a normal subgroup of group $G$

$\langle S \rangle$

the subgroup generated by $S$

$\SS_X$

the symmetric group on set $X$

$\SS_n$

the symmetric group of degree $n$

$\par \sigma$

the parity of permutation $\sigma$

$\AA_n$

the alternating group of degree $n$

$a^n$

the exponentiation of $a$ by $n$ in a ring

$\fract R$

the fraction field of integral domain $R$

$\dfrac{a}{b}$, $a / b$

elements from a fraction field

$\chara R$

the characteristic of ring $R$

$\N$

the set of natural numbers

$\N^+$

the set of positive natural numbers

$\Z$

the set of integers

$\Z^*$

the set of non-zero integers

$\Z^+$

the set of positive integers

$\Z^-$

the set of negative integers

$\Q$

the set of rational numbers

$\Q^*$

the set of non-zero rational numbers

$\Q^+$

the set of positive rational numbers

$\Q^-$

the set of negative rational numbers

$\R$

the set of real numbers

$\R^*$

the set of non-zero real numbers

$\R^+$

the set of positive real numbers

$\R^-$

the set of negative real numbers

$R[[X]]$

the ring of formal power series over ring $R$

$R[X]$

the polynomial ring over ring $R$

$\deg p$

the degree of polynomial $p$

$[\varphi]$

Iversion Brackets

$\delta_{i, j}$

Kronecker Delta

Linear Algebra

$\langle S \rangle$

the linear span of $S$

$\dim V$

the dimension of vector space $V$

$X + Y$

the sum of vector subspaces $X$ and $Y$

$X \oplus Y$

the direct sum of vector subspaces $X$ and $Y$

$\mat_R(m, n)$

the set of all $m \times n$-matrices over ring $R$

$A + B$

the sum of matrices $A$ and $B$

$AB$

the product of matrices $A$ and $B$

$\ker f$

the kernel of linear mapping $f$

$\rank f$

the rank of linear mapping $f$

$\null f$

the nullity of linear mapping $f$

Topology

$\lVert {}\cdot{} \rVert$

a seminorm

$\diam A$

the diameter of $A$

$\fun_\text{bd}$

the set of bounded functions from $X$ to $Y$

$\lVert f \rVert_\sup$

the supremum norm of $f$

$\ball_r(a)$

the open ball of radius $r$ around $a$

$\cball_r(a)$

the closed ball of radius $r$ around $a$

$\inter Y$

the interior of $Y$

Number Theory

$n \divides m$

$n$ divides $m$

Combinatorics

$n!$

the factorial of $n$

$P(n, k)$

the number of $k$-permutations of $\llbra n \rrbra$

$C(n, k)$

the number of $k$-combinations of $\llbra n \rrbra$

$\displaystyle \binom{n}{k}$

binomial coefficient [same thing as $C(n, k)$]