Formalize equivalence classes

Question

I want to formalize following:

R is equivalence relation with at least two equivalence classes.

How I can do that with only R?

Answer 1

If by 'formalize' you mean you want a logic expression:

$\exists x \exists y \ \neg R(x,y)$

This will force there to be at least two equivalence classes, since if two objects are not related through an equivalence relation $R$, then they belong to different equivalence classes ... which obviously means there are at least two equivalence classes.

Answer 2

You can say that there exist $x$ and $y$ in your set such that $x$ and $y$ are not related by the relation $R$. This is equivalent to saying that $R$ has at least two distinct equivalence classes (namely, the equivalence classes of $x$ and $y$).