# Definition:Antitransitive Relation

## Contents

## Definition

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

$\mathcal R$ is **antitransitive** if and only if:

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

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.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.5$: Properties of Relations