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.

Also see

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


 * 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 iff $f$ is a bijection, and that $f^{-1}$ is itself a bijection.


 * Left and Right Inverses of Mapping are Inverse Mapping


 * Inverse Mapping is Unique


 * Bijection Composite with Inverse, that $f^{-1}$ is the two-sided inverse of $f$, i.e.:
 * $f \circ f^{-1} = I_S$
 * $f^{-1} \circ f = I_T$

where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$.

Some sources use this property as the definition of an inverse mapping.