Definition:Transitive with Respect to a Relation

Definition
Let $A$ be a class.

Let $\mathcal R$ be a relation on $A$.

Let $S$ be a set.

Then $S$ is transitive with respect to $\mathcal R$ iff:


 * $\forall x \in A: \forall y \in S: \left({ x \mathcal R y \implies x \in S }\right)$