# Inverse of Bijection is Bijection

## Contents

## Theorem

Let $f: S \to T$ be a bijection in the sense that:

- $(1): \quad f$ is an injection
- $(2): \quad f$ is a surjection.

Then the inverse $f^{-1}$ of $f$ is itself a bijection by the same definition.

## Proof

Let $f$ be both an injection and a surjection.

From Mapping is Injection and Surjection iff Inverse is Mapping it follows that its inverse $f^{-1}$ is a mapping.

From Inverse of Inverse Relation:

- $\paren {f^{-1} }^{-1} = f$

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

But then $f$, being a bijection, is by definition a mapping.

So from Mapping is Injection and Surjection iff Inverse is Mapping it follows that $f^{-1}$ is a bijection.

$\blacksquare$

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 2$: Product sets, mappings - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 5$: Theorem $5.5$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.11$: Relations: Theorem $11.9$ - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.4$: Functions - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.3$: Mappings: Exercise $3$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.6$: Functions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\S 2$: Proposition $2.15$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.7$: Inverses: Proposition $\text{A}.7.4$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions

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