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:
 * $f = h \circ q_{\mathcal R_f}$

where:


 * $q_{\mathcal R_f}: S \to S / \mathcal R_f: \map {q_{\mathcal R_f} } s = \eqclass s {\mathcal R_f}$


 * $h: S / \mathcal R_f \to T: \map h {\eqclass s {\mathcal R_f} } = \map f s$


 * $\eqclass s {\mathcal R_f}$ denotes the equivalence class of $s$ with respect to the equivalence relation $\mathcal R$ induced on $S$ by $f$.

This can be illustrated using a commutative diagram as follows:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

S \ar[r]^*{q_{\mathcal R_f} } \ar@{-->}[rd]_*{f = h \circ q_{\mathcal R_f} } & S / \mathcal R_f \ar[d]^*{h} \\ &

T }\end{xy}$

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

The mapping $h$ is justified by Existence of Renaming Mapping.

Also see

 * Factoring Mapping into Surjection and Inclusion


 * Quotient Theorem for Surjections
 * Quotient Theorem for Sets