Diagonal Relation is Reflexive
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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.
$\blacksquare$