Factoring Mapping into Quotient and Injection

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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}\[email protected][email protected]+1em{ S \ar[r]^*{q_{\mathcal R_f} } \[email protected]{-->}[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.

$\blacksquare$


Also see


Sources