Definition:Transitivity (Relation Theory)

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

Transitive

$\RR$ is transitive if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

that is:

$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$

Antitransitive

$\RR$ is antitransitive if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \notin \RR$

that is:

$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \notin \RR$

Non-Transitive

$\RR$ is non-transitive if and only if it is neither transitive nor antitransitive.

Also see

• Reflexivity of Relations
• Symmetry of Relations

• Results about relation transitivity can be found here.