Definition:Negatively Transitive Relation

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Definition

Let $\mathcal R$ be a relation on a set $S$.


Then $\mathcal R$ is negatively transitive iff:

$\forall x, y, z \in S: \neg \left({x \mathrel{\mathcal R} y}\right) \land \neg \left({y \mathrel{\mathcal R} z}\right) \implies \neg\left({x \mathrel{\mathcal R} z}\right)$

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

$\forall x, y, z \in S: (x \mathrel{\mathcal R} z) \implies (x \mathrel{\mathcal R} y) \lor (y \mathrel{\mathcal R} z)$


Also see