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.