Bijection iff Left and Right Cancellable

Theorem

Let $f$ be a mapping.

Then $f$ is a bijection if and only if $f$ is both left cancellable and right cancellable.

Proof

Follows directly from:

Injection iff Left Cancellable: $f$ is an injection if and only if $f$ is left cancellable

Surjection iff Right Cancellable: $f$ is a surjection if and only if $f$ is right cancellable.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.9$