Let $R \subseteq A \times A$ and $S \subseteq A \times A$ be two arbitary equivalence relations. Prove or disprove that $R \cup S$ is an equivalence relation.
Reflexivity: Let $(x,x) \in R$ or $(x,x) \cup S \rightarrow (x,x) \in R \cup S$
Now I still have to prove or disprove that $R \cup S$ is symmetric and transitive. How can I do that?
My guess for symmetry is: R and S are equivalence relations, which means that $(x,y)\ and\ (y,x) \in R,S$ For each (x,y) in R and S there is an (y,x) in R and S so that (x,y) ~ (y,x). Is that correct?