Definition:Transitivity (Relation Theory)

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

Transitive
$\mathcal R$ is transitive iff:


 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Antitransitive
$\mathcal R$ is antitransitive (or intransitive) iff:


 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \notin \mathcal R$

Non-transitive
$\mathcal R$ is non-transitive iff it is neither transitive nor antitransitive.

Also see

 * Reflexivity
 * Symmetry