Definition:Antitransitive Relation

Definition

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

$\RR$ 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:

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

Also known as

Some sources use the term intransitive.

However, as intransitive is also found in other sources to mean non-transitive, it is better to use the clumsier, but less ambiguous, antitransitive.

Also see

• Definition:Transitivity (Relation Theory)

• Definition:Transitive Relation
• Definition:Non-Transitive Relation

• Results about antitransitive relations can be found here.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations