Inverse of Mapping is One-to-Many Relation

Theorem

Let $f$ be a mapping.


Then its inverse $f^{-1}$ is a one-to-many relation.

Hence $f^{-1}$ is not necessarily a mapping itself.


Proof

We have that $f$ is a mapping.

Hence $f$ is a fortiori a many-to-one relation.

Then from Inverse of Many-to-One Relation is One-to-Many, $f^{-1}$ is one-to-many.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries
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