Left Inverse Mapping is Surjection

Theorem

Let $f: S \to T$ be an injection.

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

Then $g$ is a surjection.

Proof

Let $f: S \to T$ be an injection.

Then from Injection iff Left Inverse there exists at least one left inverse $g: T \to S$ of $f$ such that $g \circ f = I_S$.

$I_S$ is a surjection.

Thus $g \circ f$ is a surjection.

By Surjection if Composite is Surjection it follows that $g$ is also a surjection.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites: Exercise $\text{(i)}$
• 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 $3$