Example of equivalence relations $S$ and $R$ such that $SR$ is an equivalence relation while $R \cup S$ isn't

Question

I need to give an example of equivalence relations $S$ and $R$ such that $SR$ is an equivalence relation while $R \cup S$ isn't. I've already tried to find such relations on small sets of natural numbers but it looks like there aren't any.

Answer 1

Consider the set $\{a,b,c,d\}$, with the equivalence relations $$\alpha = \{a,b\}^2 \cup \{c,d\}^2 \quad\text{and}\quad \beta = \{a,d\}^2 \cup \{b,c\}^2.$$ It's easy to see that $\alpha \cup \beta$ is not an equivalence relation, because it's not transitive: for example, $(a,b), (b,c) \in \alpha \cup \beta$, but $(a,c) \notin \alpha \cup \beta$.

Obviously, $\alpha \cup \beta \subseteq \alpha \circ \beta$, since the relations are reflexive.
From $$ a \equiv_{\alpha} b \equiv_{\beta} c \quad\text{and}\quad c \equiv_{\alpha} d \equiv_{\beta} a, $$ we conclude that $(a,c),(c,a) \in \alpha \circ \beta$.
Analogously we conclude that $(b,d),(d,b) \in \alpha \circ \beta$, and therefore, $\alpha \circ \beta = \{a,b,c,d\}^2$, an equivalence relation.