Left Inverse Mapping is Surjection
Jump to navigation
Jump to search
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 $3$