# Inverse Mapping is Bijection

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## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ and $g: T \to S$ be inverse mappings of each other.

Then $f$ and $g$ are bijections.

## Proof

From Inverse is Mapping implies Mapping is Injection and Surjection:

- $f$ is both an injection and a surjection.

Again from Inverse is Mapping implies Mapping is Injection and Surjection:

- $g$ is both an injection and a surjection.

The result follows by definition of bijection.

$\blacksquare$

## Also see

## Sources

- 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions