At the moment I am taking a course in logic (which includes an extensive amount of model thoery). In the last lecture we learned about quantifier elimination. On the last homework assignment we were asked the following:
Let ${\cal L}=\left\{ E\right\}$ be a binary equivalence relation symbol. For each of the following theories either prove that they have quantifier elimination or give an example showing that they do not have quantifier elimination.
a) $E$ has infinitely many equivalence classes all of size 2.
b) $E$ has infinitely many equivalence classes all of which are infinite.
c) $E$ has infinitely many classes of size 2, infinitely many classes of size 3, and every class has size 2 or 3.
d) $E$ has one equivalence class of size $n$ for each $n<\omega$ .
I'm working on the problem and I found a similar question from five years ago (here). I've gone through all the comments and linked resources, but I'm still stuck on all four sub-questions. Even with hints (that one can easily find across the links in the comments as well as similar questions), I can't even figure out if I'm supposed to prove or disprove the claims for any of them. Could someone please explain what's happening here? Answering a couple of the sub-questions would really help me develop the right approach any way of thinking.
Definition - A theory $ T$ is said to have quantifier elimination if for every formula $\varphi$ there is a quantifier free formula $\psi$ such that $T\models\varphi\iff\psi$.
