Inverse of Mapping is One-to-Many Relation

Jump to navigation Jump to search

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.

$\blacksquare$