# Quotient Theorem for Sets

## 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

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Factoring Functions: Theorem $11$