Equivalence of Definitions of Reflexive Relation

Theorem
A relation $$\mathcal R \subseteq S \times S$$ is reflexive iff it contains the diagonal relation: $$\Delta_S \subseteq \mathcal R$$.

Proof

 * Suppose $$\Delta_S \not \subseteq \mathcal R$$.

Then $$\exists \left({x, x}\right): \left({x, x}\right) \notin \mathcal R$$.

Thus $$\exists x \in S: \left({x, x}\right) \notin \mathcal R$$

and so $$\mathcal R$$ is not reflexive.


 * Now suppose $$\Delta_S \subseteq \mathcal R$$.

$$ $$

Thus $$\mathcal R$$ is reflexive.