Inverse of Bijection is Bijection
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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
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- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions: 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: Problem Set $\text{A}.4$: $25$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $3$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: 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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries
- For a video presentation of the contents of this page, visit the Khan Academy.