# Bijection iff Left and Right Cancellable

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## 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 $5.9$