Diagonal Relation is Smallest Equivalence Relation

Theorem

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.

Proof

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 $\Delta_S \subseteq \EE$.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations