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:

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}$

Also known as
Otherwise known as the factoring theorem or factor theorem.

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

Also see

 * Factoring Mapping into Quotient and Injection
 * Factoring Mapping into Surjection and Inclusion


 * Quotient Theorem for Surjections