Equivalence of Definitions of Transitive Relation

Theorem
A relation $$\mathcal R$$ is transitive iff:
 * $$\mathcal R \circ \mathcal R \subseteq \mathcal R$$

Proof

 * First, suppose $$\mathcal R$$ is transitive.

$$ $$ $$ $$


 * Now suppose $$\mathcal R$$ is not transitive. Then:

$$ $$ $$

Thus, by the Rule of Transposition, $$\mathcal R \circ \mathcal R \subseteq \mathcal R \implies \mathcal R$$ is transitive.

Comment
Some sources use this as the definition of a transitive relation.