Definition:Inverse Relation

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

The inverse (or converse) relation $$\mathcal{R}^{-1} \subseteq T \times S$$ is defined as being the relation which satisfies:


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

That is:


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

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

Note that the domain of a relation is the range of its inverse, and vice versa:


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