# Definition:Transitivity (Relation Theory)

## Definition

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

### Transitive

$\mathcal R$ is transitive if and only if:

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

that is:

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

### Antitransitive

$\mathcal R$ 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:

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

### Non-transitive

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

## Also see

• Results about relation transitivity can be found here.