# Definition:Negatively Transitive Relation

## Definition

Let $\RR$ be a relation on a set $S$.

Then $\RR$ is negatively transitive if and only if:

$\forall x, y, z \in S: \neg \paren {x \mathrel \RR y} \land \neg \paren {y \mathrel \RR z} \implies \neg \paren {x \mathrel \RR z}$

By De Morgan's Laws, this can be given the alternative form:

$\forall x, y, z \in S: \paren {x \mathrel \RR z} \implies \paren {x \mathrel \RR y} \lor \paren {y \mathrel \RR z}$