Definition:Transitive Relation/Definition 1

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is transitive iff:


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

that is:
 * $\left\{ {\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Also see

 * Equivalence of Definitions of Transitive Relation