Diagonal Relation is Smallest Equivalence Relation
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The diagonal relation $\Delta_S$ on $S$ is the smallest equivalence in $S$, in the sense that:
- $\forall \EE \subseteq S \times S: \Delta_S \subseteq \EE$
where $\EE$ denotes a general equivalence relation.
It is confirmed that, from Diagonal Relation is Equivalence, $\Delta_S$ is an equivalence relation.
Let $\EE$ be an arbitrary equivalence relation.
By definition, $\EE$ is reflexive.
From Relation Contains Diagonal Relation iff Reflexive it follows that as $\Delta_S \subseteq \EE$.
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations