Identity Mapping is a 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 an Injection
• a surjection, from Identity Mapping is a Surjection

and therefore a bijection.

$\blacksquare$

Sources

• W.E. Deskins: Abstract Algebra (1964): Exercise $1.3: \ 10$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.5$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.7$: Proposition $\text{A}.7.5 \ (1)$