Diagonal Relation is Reflexive

Theorem

The diagonal relation $\Delta_S$ on a set $S$ is a reflexive relation in $S$.


Proof

\(\ds \forall x \in S: \, \) \(\ds x\) \(=\) \(\ds x\) Definition of Equals
\(\ds \leadsto \ \ \) \(\ds \tuple {x, x}\) \(\in\) \(\ds \Delta_S\) Definition of Diagonal Relation

So $\Delta_S$ is reflexive.

$\blacksquare$

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