Definition:Antitransitive Relation

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is antitransitive iff:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \notin \mathcal R$

that is:
 * $\left\{ {\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \implies \left({x, z}\right) \notin \mathcal R$

Also known as
Some sources use the term intransitive.

However, as intransitive is also found in other sources to mean non-transitive, it is better to use the clumsier, but less ambiguous, antitransitive.

Also see

 * Definition:Transitivity (Relation Theory)


 * Definition:Transitive Relation
 * Definition:Non-transitive Relation