Proove that all equivalence relations can be split into equivalence classes Is It necessary that all equivalence relations (on any non empty set) can be divided into 2 or more equivalence classes?
If so how will we prove it?

If the relation is not trivial, that is, if not for all $x$ and $y$, $x \sim y$, then yes, there are at least two classes to relation. Let $x \not\sim y $, then define $R_x = \{a \, : \, a \sim x \}$ and $R_y=\{a\, :\, a \sim y \}$ are at least two equivalence classes. 
Why is there always exist a 'set' of all equivalence classes? I think one should use the Zorn's lemma to prove the existence of the set of all equivalence classes.
