Definition:Inverse Relation

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:

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

As $$\mathcal{R}^{-1}$$ is itself a relation, all the results that apply to relations in general will also apply to inverse relations.

Result:

The inverse of an inverse relation is the relation itself:

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

Proof:

$$\left({s, t}\right) \in \mathcal{R}$$

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

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

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