Let R be an equivalence relation defined on a set A. For any x, y ∈ A, either [x] ∩ [y] = ∅ or [x] = [y]. (Prove or Disprove

Question

Let R be an equivalence relation defined on a set A. Prove or Disprove "For any x, y ∈ A, either [x] ∩ [y] = ∅ or [x] = [y]."

Having a bit of trouble with this question because I'm not sure how to utilize the equivalence nature of the relation to the intersection.

Answer 1

Suppose $[x]\cap[y]$ is not empty. Then there is an element $z$ such that $zRx$ and $zRy$.

But then, by the usual properties of equivalence relations, you obtain $xRy$ and therefore $[x]=[y]$.