Quotient Theorem for Sets/Proof

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Theorem

A mapping $f: S \to T$ can be uniquely factored into a surjection, followed by a bijection, followed by an injection.

Thus:

$f = i \circ r \circ q_{\RR_f}$

where:

\(\ds q_{\RR_f}: \ \ \) \(\, \ds S \to S / \RR_f: \, \) \(\ds \map {q_{\RR_f} } s\) \(=\) \(\ds \eqclass s {\RR_f}\) Quotient Mapping
\(\ds r: \ \ \) \(\, \ds S / \RR_f \to \Img f: \, \) \(\ds \map r {\eqclass s {\RR_f} }\) \(=\) \(\ds \map f s\) Renaming Mapping
\(\ds i: \ \ \) \(\, \ds \Img f \to T: \, \) \(\ds \map i t\) \(=\) \(\ds t\) Inclusion Mapping

where:

$\RR_f$ is the equivalence induced by $f$
$S / \RR_f$ is the quotient set of $S$ induced by $\RR_f$


This can be illustrated using a commutative diagram as follows:

$\begin {xy} \xymatrix@L + 2mu@ + 1em { S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} } \ar[d]_*{q_{\RR_f} } & & & T \\ S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[u]_*{i_T} } \end {xy}$


Proof

From Factoring Mapping into Surjection and Inclusion, $f$ can be factored uniquely into:

A surjection $g: S \to \Img f$, followed by:
The inclusion mapping $i: \Img f \to T$ (an injection).


$\begin{xy}\xymatrix@L+2mu@+1em { S \ar[drdr]_*{g} \ar@{-->}[rr]^*{f = i_T \circ g} & & T \\ \\ & & \Img f \ar[uu]_*{i_T} }\end{xy}$


From the Quotient Theorem for Surjections, the surjection $g$ can be factored uniquely into:

The quotient mapping $q_{\RR_f}: S \to S / \RR_f$ (a surjection), followed by:
The renaming mapping $r: S / \RR_f \to \Img f$ (a bijection).


Thus:

$f = i_T \circ \paren {r \circ q_{\RR_f} }$

As Composition of Mappings is Associative it can be seen that $f = i_T \circ r \circ q_{\RR_f}$.


The commutative diagram is as follows:

$\begin {xy} \xymatrix@L + 2mu@ + 1em { S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} } \ar[ddd]_*{q_{\RR_f} } \ar@{..>}[drdrdr]_*{g = r \circ q_{\RR_f} } & & & T \\ \\ \\ S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[uuu]_*{i_T} } \end {xy}$

$\blacksquare$


Also known as

Otherwise known as the factoring theorem or factor theorem.


This construction is known as the canonical decomposition of $f$.


Also see


Sources