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.