Universal Property of Quotient Set
Theorem
Let $X$ and $Y$ be sets.
Let $\sim$ be an equivalence relation on $X$.
Let $\pi : X \to X/\sim$ be the quotient mapping to the quotient set.
Let $f : X \to Y$ be $\sim$-invariant.
Then there exists a unique mapping $\overline f : X/\!\sim \to Y$ such that $f = \overline f \circ \pi$.
- $\xymatrix{
X \ar[d]_\pi \ar[r]^f & Y\\ X/\sim \ar@{.>}[ru]_{\overline f} }$
Proof
Existence
For $x \in X$, let $[x] \in X/\sim$ denote its equivalence class under $\sim$.
Define the relation $\overline f : X/\sim \to Y$ by:
- $([x], y) \in \overline f \iff f(x) = y$.
We check that this is a well-defined mapping.
By Quotient Mapping is Surjection, $f$ is left-total.
Let $[x] \in X/\sim$ and $y, z \in Y$ with:
- $([x], y) \in \overline f$
- $([x], z) \in \overline f$
We show that $y = z$.
By definition of $\overline f$, there exist $u, v \in X$ with $[u] = [v] = [x]$ and $f(u) = y$ and $f(v) = z$.
By definition of equivalence class, $u \sim v$.
Because $f$ is $\sim$-invariant, $y = z$.
Thus $\overline f$ is many-to-one.
Thus $\overline f$ is a well-defined mapping.
$\Box$
Uniqueness
Follows from:
$\blacksquare$