Bijection iff Left and Right Inverse

Theorem

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

$f$ is a bijection if and only if:

$\text {(1)}: \quad$

$\ds \exists g_1: T \to S: \, $

$\ds g_1 \circ f$

$=$

$\ds I_S$

$\text {(2)}: \quad$

$\ds \exists g_2: T \to S: \, $

$\ds f \circ g_2$

$=$

$\ds I_T$

where:

$g_1$ and $g_2$ are mappings

$\circ$ denotes composition of mappings.

It also follows that it is necessarily the case that $g_1 = g_2$ for such to be possible.

Corollary

Let $f: S \to T$ and $g: T \to S$ be mappings such that:

$\ds g \circ f$

$=$

$\ds I_S$

$\ds f \circ g$

$=$

$\ds I_T$

Then both $f$ and $g$ are bijections.

Proof

Necessary Condition

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

Let $f$ be such that:

$\ds \exists g_1: T \to S: \, $

$\ds g_1 \circ f$

$=$

$\ds I_S$

$\ds \exists g_2: T \to S: \, $

$\ds f \circ g_2$

$=$

$\ds I_T$

where both $g_1$ and $g_2$ are mappings.

Then from Left and Right Inverse Mappings Implies Bijection it follows that $f$ is a bijection.

From Left and Right Inverses of Mapping are Inverse Mapping it follows that:

$g_1 = g_2 = f^{-1}$

where $f^{-1}$ is the inverse of $f$.

Sufficient Condition

Let $f: S \to T$ be a bijection.

Then from Bijection has Left and Right Inverse it follows that:

$f^{-1} \circ f = I_S$

and:

$f \circ f^{-1} = I_T$

where $f^{-1}$ is the inverse of $f$.

Also see

Composite of Bijection with Inverse is Identity Mapping for the converse of this result.

Sources

1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions: Theorem $9$
1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Inverse images and inverse functions
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bijection
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Lemma $2.1$
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.5 \ \text{(e)}$
2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Bijections
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bijection