# Factoring Mapping into Quotient and Injection

## Theorem

Let $f: S \to T$ be a mapping.

Then $f$ can be uniquely factored into a quotient mapping, followed by an injection.

Thus:

$f = h \circ q_{\RR_f}$

where:

$q_{\RR_f}: S \to S / \RR_f: \map {q_{\RR_f} } s = \eqclass s {\RR_f}$
$h: S / \RR_f \to T: \map h {\eqclass s {\RR_f} } = \map f s$
$\eqclass s {\RR_f}$ denotes the equivalence class of $s$ with respect to the equivalence relation $\RR$ induced on $S$ by $f$.

This can be illustrated using a commutative diagram as follows:

$\begin{xy}\[email protected][email protected]+1em{ S \ar[r]^*{q_{\RR_f} } \[email protected]{-->}[rd]_*{f = h \circ q_{\RR_f} } & S / \RR_f \ar[d]^*{h} \\ & T }\end{xy}$

## Proof

The mapping $q_{\RR_f}: S \to S / \RR_f$ follows from the definition of quotient mapping.

The mapping $h$ is justified by Condition for Mapping from Quotient Set to be Well-Defined.

$\blacksquare$