Quotient set

Let $\sim$ be an equivalence relation on a set $A$. The quotient set $A/{\sim}$ is the set of all equivalence classes :

There is a canonical surjective function $\pi\colon A\to A/{\sim}$, called the quotient map, defined by $\pi(a)=[a]$.

Examples:

• For congruence modulo $n$ on $\mathbb{Z}$, the quotient set $\mathbb{Z}/{\sim}$ is the set of residue classes modulo $n$ (often written $\mathbb{Z}/n\mathbb{Z}$ as a set).
• If $\sim$ is equality on $A$, then $A/{\sim}$ can be identified with $A$ via $a\mapsto [a]=\{a\}$.