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 $\RR$ be an equivalence relation on $V$ such that:
 * $S \subseteq \RR$

Then $\RR$ is the trivial relation on $S$.

Proof
By definition, $\RR$ is connected.

We also have that $\RR$ is an equivalence relation.

The result follows from Connected Equivalence Relation is Trivial.