Definition:Transitivity (Relation Theory)

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

Transitive
$$\mathcal R$$ is transitive iff:


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

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


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

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

Also see

 * Reflexivity
 * Symmetry