Definition:Inverse Relation

Definition
Let $$\mathcal R \subseteq S \times T$$ be a relation.

The inverse (or converse) relation to (or of) $$\mathcal R$$ is defined as:


 * $$\mathcal R^{-1} \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$$

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


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

Domain and range of Inverse Relation
Note that the preimage of a relation is the image of its inverse, and vice versa:


 * $$\operatorname {Im}^{-1} \left({\mathcal R}\right) = \operatorname {Im} \left({\mathcal R^{-1}}\right)$$
 * $$\operatorname {Im} \left({\mathcal R}\right) = \operatorname {Im}^{-1} \left({\mathcal R^{-1}}\right)$$

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

Others, for example, use $$\mathcal R^t$$.

Also see

 * Inverse of a Mapping
 * Inverse Mapping