Equivalence of Definitions of Transitive Relation

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

where $\circ$ denotes composite relation.

Necessary Condition
First, suppose $\mathcal R$ is transitive.

Sufficient Condition
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.