Surjection iff Right Inverse

Theorem
A mapping $f: S \to T, S \ne \varnothing$ is a surjection iff:
 * $\exists g: T \to S: f \circ g = I_T$

where:
 * $g$ is a mapping
 * $I_T$ is the identity mapping on $T$.

That is, if $f$ has a right inverse.

In general, that right inverse is not unique.

Uniqueness occurs iff $f$ is an injection.

Proof of Non-Uniqueness
If $f$ is not an injection then:
 * $\exists y \in T: \exists x_1, x_2 \in S: f \left({x_1}\right) = y = f \left({x_2}\right)$

Hence we have more than one choice in $f^{-1}\left({\left\{{y}\right\}}\right)$ for how to map $g \left({y}\right)$.

This does not happen iff $f$ is an injection.

Hence the result.

Also see

 * Injection iff Left Inverse