Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$

A binary relation on a set $A$ is a subset $R\subseteq A\times A$ (Relation, $\mathrm{dom}R$, $\mathrm{ran}R$, $\mathrm{fld}R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$", The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$); we write $a\sim b$ for $(a,b)\in R$ and read it "$a$ is related to $b$".

A binary relation $\sim$ on $A$ is an equivalence relation when it is reflexive on $A$, symmetric and transitive (Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, and connex relations on a set), that is, when it is

• reflexive: $a\sim a$ for every $a\in A$;
• symmetric: $a\sim b$ implies $b\sim a$, for all $a,b\in A$;
• transitive: $a\sim b$ and $b\sim c$ imply $a\sim c$, for all $a,b,c\in A$.

Let $\sim$ be an equivalence relation on $A$. For $a\in A$, the equivalence class of $a$ is the subset

$[a]:=\{b\in A:a\sim b\}\subseteq A,$

and any $b$ with $[b]=C$ is called a representative of the class $C$. The quotient set is the set of all classes,

$A\mathrm{/}\sim :=\{[a]:a\in A\},$

a subset of the power set of $A$ (The power set $\mathcal{P}(x)=\{z:z\subseteq x\}$). The map $\pi :A\to A\mathrm{/}\sim$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) with $\pi (a)=[a]$ is the quotient map; it is surjective by construction (Injection, surjection, bijection).

• What the classes look like is the content of The equivalence classes of an equivalence relation are nonempty, cover $A$, and are pairwise equal or disjoint; conversely every such cover arises from exactly one equivalence relation: they are nonempty, they cover $A$, and any two of them are either equal or disjoint. That lemma also runs the converse, so "equivalence relation on $A$" and "partition of $A$" are two names for the same data.

• Well-definedness. A rule that assigns a value to $[a]$ by a formula in the representative $a$ defines a function on $A\mathrm{/}\sim$ only if the value is the same for every representative, that is, only if $a\sim a^{\mathrm{\prime }}$ implies that the formula returns the same value at $a$ and at $a^{\mathrm{\prime }}$. The general statement of that obligation, with the uniqueness of the induced function, is Let $\sim$ be an equivalence relation on $A$ with quotient map $\pi :A\to A\mathrm{/}\sim$, and let $f:A\to B$. There is a function $g:A\mathrm{/}\sim \to B$ with $g\circ \pi =f$ if and only if $a\sim a^{\mathrm{\prime }}$ implies $f(a)=f(a^{\mathrm{\prime }})$; and such a $g$ is then unique. The constructions of $\mathbb{Z}$ and of $\mathbb{Q}$ discharge it for their own operations (Integer addition and negation are well defined ↗, Integer multiplication is well defined ↗, Rational arithmetic is well defined ↗).

• Where the general notion is used. The construction of $\mathbb{Z}$ verifies the three properties for the concrete relation $(a,b)\sim (c,d)⟺a+d=b+c$ (The integer-defining relation is an equivalence relation ↗), and the construction of $\mathbb{Q}$ does the same for its own relation (The rational-defining relation is an equivalence relation ↗). Each of those items settles one relation and defines no general notion. Cosets, quotient groups, congruence classes and quotient rings are further instances of the definition given here.