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.

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

$$ $$ $$

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