# Bijection iff Left and Right Inverse

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## Contents

## Theorem

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

$f$ is a bijection if and only if:

- $(1)\quad$ $\exists g_1: T \to S: g_1 \circ f = I_S$
- $(2)\quad$ $\exists g_2: T \to S: f \circ g_2 = I_T$

where both $g_1$ and $g_2$ are 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:

- $g \circ f = I_S$
- $f \circ g = I_T$

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

## Proof

### Necessary Condition

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

Let $f$ be such that:

- $\exists g_1: T \to S: g_1 \circ f = I_S$
- $\exists g_2: T \to S: f \circ g_2 = I_T$

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 - 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