Definition:Transitive Relation/Definition 1

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


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

that is:
 * $\set {tuple {x, y}, tuple {y, z} } \subseteq \mathcal R \implies tuple {x, z} \in \mathcal R$

Also see

 * Equivalence of Definitions of Transitive Relation