Equivalence of Definitions of Dual Relation

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

Then the following definitions of the dual of $\mathcal R$ are equivalent:

Proof
Let $\left({x, y}\right) \in \left({\overline{\mathcal R}}\right)^{-1}$.

Then we reason as follows: