= Canonical projection of a quotient set
{title2=$q:X\to X/{\sim},\quad q(x)=[x]$}
= Quotient set projection
{synonym}
The canonical projection of a quotient set is the <function> $q(x)=[x]$. It is <surjective>: every <set element> of $X/{\sim}$ is $[x]$ for some $x\in X$. Two <set elements> have the same image exactly when they are related by the <equivalence relation>. Its <fibers> are precisely the <equivalence classes>:
$$
q^{-1}(\{[x]\})=[x].
$$
Here the left side is the <preimage> of a <singleton> <subset> of the <quotient set>; its members are the original <set elements> in the class $[x]$. In the diagram, the three arrows arriving at $[1]$ start at $1$, $4$ and $7$.
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