Quotient set

Definition

Let $\sim$ be an equivalence relation on a set $A$. The quotient set of $A$ by $\sim$ is the set $$A/{\sim} = \{[a] \mid a \in A\}$$ of all equivalence classes. The canonical projection is the function $\pi \colon A \to A/{\sim}$ with $\pi(a) = [a]$.

Remarks

For congruence modulo $n$ on $\mathbb{Z}$, the quotient set $\mathbb{Z}/n\mathbb{Z}$ has the $n$ elements $[0], [1], \dots, [n-1]$. The canonical projection is surjective. A function on $A/{\sim}$ is often defined by a formula on representatives. The formula must give the same value for equivalent representatives.

Explanations

L1 human

In the quotient set, each class becomes a single point. From $\mathbb{Z}$ with congruence modulo $2$ you get a set with two elements: "even" and "odd".