Category:Transitive Relations

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
$\RR$ is transitive if and only if:
$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$
$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$