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:
\(\ds q_{\RR_f}: \ \ \) | \(\, \ds S \to S / \RR_f: \, \) | \(\ds \map {q_{\RR_f} } s\) | \(=\) | \(\ds \eqclass s {\RR_f}\) | 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:
$\quad\quad \begin {xy} \xymatrix@L + 2mu@ + 1em { S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} } \ar[d]_*{q_{\RR_f} } & & & T \\ S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[u]_*{i_T} } \end {xy}$
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).
$\quad\quad \begin{xy}\xymatrix@L+2mu@+1em {
S \ar[drdr]_*{g}
\ar@{-->}[rr]^*{f = i_T \circ g}
& & T
\\ \\
& & \Img f \ar[uu]_*{i_T}
}\end{xy}$
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:
$\quad\quad \begin {xy} \xymatrix@L + 2mu@ + 1em { S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} } \ar[ddd]_*{q_{\RR_f} } \ar@{..>}[drdrdr]_*{g = r \circ q_{\RR_f} } & & & T \\ \\ \\ S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[uuu]_*{i_T} } \end {xy}$
$\blacksquare$
Also known as
Otherwise known as the factoring theorem or factor theorem.
This construction is known as the canonical decomposition of $f$.
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 $\RR$:
\(\ds \R / \RR_f\) | \(:=\) | \(\ds \set {\eqclass x {\RR_f}: x \in \R}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\set {x, -x}: x \in \R}\) |
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 $11$