Injection iff Left Inverse

Theorem
A mapping $f: S \to T, S \ne \varnothing$ is an injection iff:
 * $\exists g: T \to S: g \circ f = I_S$

where $g$ is a mapping.

That is, iff $f$ has a left inverse.

In general, that left inverse is not unique.

Uniqueness occurs under either of two circumstances:


 * $S$ is a singleton
 * $f$ is a bijection.

Also see

 * Surjection iff Right Inverse