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.

$\blacksquare$


Also see


Sources

  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
  • 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 8$: Composition of Functions and Diagrams: Exercise $3$
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