Definition:Inverse of Mapping

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

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

The inverse of $f$ is the relation defined as:
 * $f^{-1} := \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

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

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


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

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

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

Also see

 * Definition:Inverse Relation
 * Definition:Inverse Mapping


 * Definition:Preimage of Mapping (also known as inverse image)


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