Definition:Transitivity (Relation Theory)

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

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

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

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

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

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