Union of Equivalences/Proof 1

Proof
This can be shown by giving an example.

Let $S = \set {a, b, c}$, and let $\RR_1$ and $\RR_2$ be equivalences on $S$ such that:


 * $\eqclass a {\RR_1} = \eqclass b {\RR_1} = \set {a, b}$
 * $\eqclass c {\RR_1} = \set c$
 * $\eqclass a {\RR_2} = \set a$
 * $\eqclass b {\RR_2} = \eqclass c {\RR_2} = \set {b, c}$

Let $\RR_3 = \RR_1 \cup \RR_2$.

Then:

However:

So $\RR_3$ is not transitive, and therefore $\RR_3 = \RR_1 \cup \RR_2$ is not an equivalence relation.