# Definition:Inverse of Mapping

Not to be confused with Definition:Inverse Mapping.

## Definition

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping

The inverse of $f$ is its inverse relation, defined as:

$f^{-1} := \set {\tuple {t, s}: \map f s = t}$

That is:

$f^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in f}$

That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies:

$\forall s \in S: \forall t \in T: \tuple {t, s} \in f^{-1} \iff \tuple {s, t} \in f$

## Also known as

The inverse of a mapping is also known as its converse.

## Also denoted as

Some authors use the notation $f^\gets$ or $f^{\circ-1}$ instead of $f^{-1}$.