Relation on Empty Set is Equivalence

Theorem

Let $S = \O$, that is, the empty set.

Let $\RR \subseteq S \times S$ be a relation on $S$.

Then $\RR$ is the null relation and is an equivalence relation.

Proof

As $S = \O$, we have from Cartesian Product is Empty iff Factor is Empty that $S \times S = \O$.

Then it follows that $\RR \subseteq S \times S = \O$.

Reflexivity

From the definition:

$\RR = \O \implies \forall x \in S: \tuple {x, x} \notin \RR$

But as $\neg \, \exists x \in S$ it follows vacuously that $\RR$ is reflexive.

$\Box$

Symmetry

It follows vacuously that:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

and so $\RR$ is symmetric.

$\Box$

Transitivity

It follows vacuously that:

$\tuple {x, y}, \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

and so $\RR$ is transitive.

$\Box$

It follows from the definition that $\RR$ is an equivalence relation.

$\blacksquare$