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I want to formalize following:

R is equivalence relation with at least two equivalence classes.

How I can do that with only R?

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  • $\begingroup$ Do what? I think it's unclear what you're asking about. $\endgroup$
    – skyking
    May 8 '17 at 7:34
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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.

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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 $y$).

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