Inverse Mapping is Bijection

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
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