Connected Equivalence Relation is Trivial/Examples/Arbitrary Set of 4 Elements

Example of Connected Equivalence Relation is Trivial

Let $V = \set {a, b, c, d}$.

Let $S \subseteq V \times V$ such that:

$S = \set {\tuple {a, b}, \tuple {b, c}, \tuple {c, d} }$

Let $\mathcal R$ be an equivalence relation on $V$ such that:

$S \subseteq \mathcal R$

Then $\mathcal R$ is the trivial relation on $S$.

Proof

By definition, $\mathcal R$ is connected.

We also have that $\mathcal R$ is an equivalence relation.

The result follows from Connected Equivalence Relation is Trivial.

$\blacksquare$