Definition:Inverse Mapping

Definition
Let $S$ and $T$ be sets.

Also known as
If $f$ has an inverse mapping, then $f$ is an invertible mapping.

When $f^{-1}$ is a mapping, we say that $f$ has an inverse mapping.

Some sources, in distinguishing this from a left inverse and a right inverse, refer to this as the two-sided inverse.

Some sources use the term converse mapping for inverse mapping.

If $g$ is an inverse of $f$, then $f$ and $g$ are also called mutually inverse bijections, inverse bijections or reverse bijections.

Also defined as
Some authors gloss over the fact that $f$ needs to be a surjection for the inverse of $f$ to be a mapping:

Such is the approach of.

Also see

 * Equivalence of Definitions of Inverse Mapping (use Composite of Bijection with Inverse is Identity Mapping)


 * Bijection iff Left and Right Inverse, which demonstrates that if $f$ and $f^{-1}$ are inverse mappings, they are both bijections.


 * Bijection iff Inverse is Bijection, where is shown that $f^{-1}$ is a mapping $f$ is a bijection, and that $f^{-1}$ is itself a bijection.


 * Left and Right Inverses of Mapping are Inverse Mapping


 * Inverse Mapping is Unique