Right Inverse Mapping is Injection

Theorem
Let $f: S \to T$ be a mapping.

Let $g: T \to S$ be a right inverse of $f$.

Then $g$ is an injection.

Proof
By the definition of right inverse:


 * $f \circ g = I_T$

where $I_T$ is the identity mapping on $T$.

By Identity Mapping is Injection, $I_T$ is an injection.

By Injection if Composite is Injection, it follows that $g$ is an injection.

Also see

 * Left Inverse Mapping is Surjection