Equivalence of Definitions of Transitive Relation

Definition 1 implies Definition 2
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Then:

Thus $\mathcal R$ is transitive by definition 2.

Definition 2 implies Definition 1
Let $\mathcal R$ be a relation.

We use a Proof by Contraposition by showing that


 * $\neg\left(\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R\right)\implies \mathcal R \circ \mathcal R\not \subseteq\mathcal R$.

Thus, suppose $\mathcal R$ does not fulfil the condition:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Then:

From Rule of Transposition it follows that


 * $\mathcal R \circ \mathcal R\not \subseteq\mathcal R \implies \left(\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R\right)$.