Definition:Inverse of Mapping

Definition
The inverse (or converse) of a mapping $$f: S \to T$$ is the relation defined as:
 * $$f^{-1} \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$$

and can be alternatively be defined:
 * $$f^{-1} \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{\left({t, s}\right): \left({s, t}\right) \in f}\right\}$$

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.

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

Also see

 * Inverse Relation
 * Inverse Mapping


 * Inverse Image (also known as Preimage)