Definition:Complement of Relation

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

The complement of $$\mathcal{R}$$ is defined as:

$$\mathcal{C} \left({\mathcal{R}}\right) = \left\{{\left({s, t}\right) \in S \times T: \left({s, t}\right) \notin \mathcal{R}}\right\}$$

An alternative to $$\mathcal{C} \left({\mathcal{R}}\right)$$ is $$\overline{\mathcal{R}}$$ which is more compact and convenient, but the context needs to be established so that it does not get confused with other usages of the overline notation.

Some authors use $$\mathcal{R}'$$ to denote the complement of $$\mathcal{R}$$, but $$'$$ is already heavily overused.