Diagonal Relation is Symmetric
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Theorem
The diagonal relation $\Delta_S$ on a set $S$ is a symmetric relation in $S$.
Proof
| \(\ds \forall x, y \in S: \, \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \Delta_S\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds y\) | Definition of Diagonal Relation | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds x\) | Equality is Symmetric | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {y, x}\) | \(\in\) | \(\ds \Delta_S\) | Definition of Diagonal Relation |
So $\Delta_S$ is symmetric.
$\blacksquare$