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 surjection.

Bijection has Unique Left Inverse
If $f$ is a surjection as well as an injection (and thus a bijection, then $T \setminus \operatorname{Im} \left({f}\right) = \varnothing$, and we have that $g = f^{-1}$.

As $f^{-1}$ is uniquely defined $g$ is itself unique.

Injection from Singleton has Unique Left Inverse
If $S$ is a singleton then there can only be one mapping $g: T \to S$:
 * $\forall t \in T: g \left({t}\right) = s$

Left Inverse is not Unique in Other Cases
If $f$ is not a surjection, then $T \setminus \operatorname{Im} \left({f}\right) \ne \varnothing$.

Let $t \in T \setminus \operatorname{Im} \left({f}\right)$.

We can now choose any $x_0 \in S$ such that $g \left({t}\right) = x_0$.

If $S$ is not a singleton, such an $x_0$ is not unique.

Hence the result.

Also see

 * Surjection iff Right Inverse