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.