This is not a proof. I'll give you an outline or a proof. you want to prove (i) that all element in S belong to an equivalence class and (ii) that any two equivalence classes are either disjoint or equal and that once you have proved these two properties your pretty much done.

i) is trivial using the reflexive property.

ii) first suppose and share and element x. then by definition and using the transitivity and symmetry properties you can show that . I'll leave you to prove that iff .

Bobak