Quotient Theorem for Sets

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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:

\(\displaystyle q_{\RR_f}: \ \ \) \(\, \displaystyle S \to S / \RR_f: \, \) \(\displaystyle \map {q_{\RR_f} } s\) \(=\) \(\displaystyle \eqclass s {\RR_f}\) Quotient Mapping
\(\displaystyle r: \ \ \) \(\, \displaystyle S / \RR_f \to \Img f: \, \) \(\displaystyle \map r {\eqclass s {\RR_f} }\) \(=\) \(\displaystyle \map f s\) Renaming Mapping
\(\displaystyle i: \ \ \) \(\, \displaystyle \Img f \to T: \, \) \(\displaystyle \map i t\) \(=\) \(\displaystyle t\) 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} \[email protected] + [email protected] + 1em { S \[email protected]{-->}[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}\[email protected][email protected]+1em { S \ar[drdr]_*{g} \[email protected]{-->}[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} \[email protected] + [email protected] + 1em { S \[email protected]{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} } \ar[ddd]_*{q_{\RR_f} } \[email protected]{..>}[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 $f$.


Examples

Real Square Function

Let $f: \R \to \R$ denote the square function:

$\forall x \in \R: \map f x = x^2$


We define $\RR_f \subseteq S \times S$ to be the relation:

$\tuple {x_1, x_2} \in \RR_f \iff {x_1}^2 = {x_2}^2$

that is:

$x_1 \mathrel {\RR_f} x_2 \iff x_1 = \pm x_2$


The quotient set of $\R$ induced by $\RR_f$ is thus the set $\R / \RR_f$ of $\RR$-classes of $\RR$:

\(\displaystyle \R / \RR_f\) \(:=\) \(\displaystyle \set {\eqclass x {\RR_f}: x \in \R}\)
\(\displaystyle \) \(=\) \(\displaystyle \set {\set {x, -x}: x \in \R}\)


Hence the quotient mapping $q_{\RR_f}$:

$q_{\RR_f}: \R \to \R / \RR_f: \map {q_{\RR_f} } x = \eqclass x {\RR_f}$

puts $x$ into its equivalence class $\set {x, -x}$.

We note in passing that $\eqclass x {\RR_f}$ has $2$ elements unless $x = 0$.


The renaming mapping is defined as:

$r: \R / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = x^2$

where $\Img f = \R_{\ge 0}$.


Finally the inclusion mapping is defined as:

$i_{\R_{\ge 0} }: \R_{\ge 0} \to \R: \map {i_{\R_{\ge 0} } } x = x$


Also see


Sources