Reflexive Closure is Reflexive

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Theorem

Let $\mathcal R$ be a relation on a set $S$.

Then $\mathcal R^=$, the reflexive closure of $\mathcal R$, is reflexive.


Proof

Recall the definition of reflexive closure:

$\mathcal R^= := \mathcal R \cup \Delta_S$

From Set is Subset of Union:

$\Delta_S \subseteq \mathcal R^=$

The result follows directly from Relation Contains Diagonal Relation iff Reflexive.

$\blacksquare$


Sources