Definition:Non-Transitive Relation

Definition

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

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

Also known as

Some sources use the term intransitive relation for a non-transitive relation.

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

Also see

• Definition:Transitivity (Relation Theory)

• Definition:Transitive Relation
• Definition:Antitransitive Relation

• Results about non-transitive relations can be found here.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): non-transitive (of a relation)