Connected Equivalence Relation is Trivial/Examples/Arbitrary Set of 4 Elements
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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.
$\blacksquare$
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $10$