Composite of Bijections

Theorem

Every composite of bijections is also a bijection.

That is:

If $f$ and $g$ are both bijections, then so is $f \circ g$.

Proof

• As every bijection is also by definition an injection, a composite of bijections is also a composite of injections.

Every composite of injections is also an injection by Composite of Injections is an Injection.

• As every bijection is also by definition a surjection, a composite of bijections is also a composite of surjections.

Every composite of surjections is also a surjection by Composite of Surjections is a Surjection.

• As a composite of bijections is therefore both an injection and a surjection, it is also a bijection.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.6$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Exercise $\text{J}$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$: Theorem $5.10 \ (3)$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 25.2$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Corollary $2.16$