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.
$\blacksquare$
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