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

A binary relation on a set $A$ is a subset $R \subseteq A \times A$ (Relation, $\operatorname{dom} R$, $\operatorname{ran} R$, $\operatorname{fld} R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$", The Cartesian product $A \times B := \{\, z \in \mathcal{P}(\mathcal{P}(A \cup B)) : \exists a \in A\ \exists b \in B\ z = (a,b) \,\}$); 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,

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

a subset of the power set of $A$ (The power set $\mathcal{P}(x) = \{\, z : z \subseteq x \,\}$). The map $\pi : A \to A/{\sim}$ (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 $A/{\sim}$ only if the value is the same for every representative, that is, only if $a \sim a'$ implies that the formula returns the same value at $a$ and at $a'$. 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 $\pi : A \to A/{\sim}$, and let $f : A \to B$. There is a function $g : A/{\sim} \to B$ with $g \circ \pi = f$ if and only if $a \sim a'$ implies $f(a) = f(a')$; 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 $(a,b) \sim (c,d) \iff a + d = b + c$ (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.