Definition:Transitivity (Relation Theory)

Definition

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

Transitive

$\mathcal R$ is transitive iff:

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

that is:

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

Antitransitive

$\mathcal R$ 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:

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

Non-transitive

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

Also see

• Reflexivity of Relations
• Symmetry of Relations

• Results about relation transitivity can be found here.