Diagonal Relation is Smallest Equivalence Relation

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Theorem

The diagonal relation $\Delta_S$ on $S$ is the smallest equivalence in $S$, in the sense that:

$\forall \mathcal E \subseteq S \times S: \Delta_S \subseteq \mathcal E$

where $\mathcal E$ denotes a general equivalence relation.


Proof

It is confirmed that, from Diagonal Relation is Equivalence, $\Delta_S$ is an equivalence relation.

Let $\mathcal E$ be an arbitrary equivalence relation.

By definition, $\mathcal E$ is reflexive.

From Relation Contains Diagonal Relation iff Reflexive it follows that as $\Delta_S \subseteq \mathcal E$.

$\blacksquare$


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