The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$

Fix an integer $n$. Congruence modulo $n$ is an equivalence relation on $\mathbb{Z}$ by Congruence modulo every integer is an equivalence relation on $\mathbb{Z}$. The congruence class of $a$ modulo $n$ is

$[a{]}_n:=\{b\in \mathbb{Z}:b\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}n)\},$

and the integers modulo $n$ form the quotient set

$\mathbb{Z}\mathrm{/}n:=\{[a{]}_n:a\in \mathbb{Z}\}\mathrm{.}$

This is the quotient-set construction of Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$. By 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, two classes are equal exactly when their representatives are congruent:

$[a{]}_n=[b{]}_n⟺a\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}n)\mathrm{.}$

At $n=0$ each class is a singleton because congruence modulo $0$ is equality. At $n=1$ there is one class, namely $\mathbb{Z}$ itself.

• The notation $\mathbb{Z}\mathrm{/}n$ in this item denotes a quotient set. It does not yet assert any algebraic structure.
• Since congruence modulo $n$ and modulo $-n$ are the same relation, their quotient sets are literally the same collection of subsets of $\mathbb{Z}$.