Suppose we have two equivalence relations $A$, $B$ s. t. $|A|=|B|$ (equipotent sets).
Is $A$ elementarily equivalent to $B$, $A \equiv B$?
My definition of elementarily equivalent is the following: $A \equiv B$ iff $Th(A)=Th(B)$.
I've tried to use the definition taking an arbitrary sentence $\varphi$ in the lexicon $\mathcal{L}$ s. t. $A \models \varphi$ or $B \models \varphi$ , also, I tried to go through equivalence classes, but I don't even know how to proceed.
I would appreciate some hint/answer, Thanks in advance!
Best Answer
If you have two equivalence relations with the same finite number of classes, they'll be elementarily equivalent if and only if for each possible size of class (an element of $\mathbb{N} \cup \{\infty\}$, we don't distinguish infinite cardinals) they have the same number of classes with that size.
e.g say $A$ has two classes, one with 2 elements, the other with infinitely many elements. Say $B$ has two classes, one with 3 elements, the other with infinitely many elements. Then $A$ and $B$ are not elementary equivalent since the formula $\varphi := \exists x \exists y \ \Big(x \neq y \wedge \forall z\ \ (xEz \to (z = y \vee z = x)\big)\Big)$ holds in $A$, but not in $B$.