This is a textbook problem, and I'm trying to understand how it can be proved. I feel like I can come up with a counterexample, but assume I must be wrong somewhere in my understanding.
For sake of counterexample, assume A = {1,2}, R={(1,1)}, S={(2,2)}, then R and S are equivalence relations (as they are both reflexive, symmetric and transitive). However, the intersection of R and S is the empty set. Since an equivalence relation cannot be the empty set, then $R\cap S$ is not an equivalence relation.
So... what is my mistake here?
$R$ and $S$ are not equivalence relations. $(2,2)\not\in R$ and $(1,1)\not\in S$.