# Definition:Antitransitive Relation

## Definition

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

$\mathcal R$ 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:

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

## 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

• Results about relation transitivity can be found here.