Definition:Inverse of Mapping

Definition

The inverse (or converse) of a mapping $f: S \to T$ is the relation defined as:

$f^{-1} := \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

and can be alternatively be defined:

$f^{-1} := \left\{{\left({t, s}\right): \left({s, t}\right) \in f}\right\}$

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

$\forall s \in S: \forall t \in T: \left({t, s}\right) \in f^{-1} \iff \left({s, t}\right) \in f$

From Inverse of Mapping is One-to-Many Relation, it it clear that $f^{-1}$ is in general not itself a mapping.

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