Identity Mapping is Bijection

Theorem

The identity mapping $I_S: S \to S$ on the set $S$ is a bijection.

Proof

The identity mapping is:

an injection, from Identity Mapping is Injection

a surjection, from Identity Mapping is Surjection

and therefore a bijection.

$\blacksquare$

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.3: \ 10$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.5$. Identity mappings
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.7$: Inverses: Proposition $\text{A}.7.5 \ (1)$