Full question: Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation and let $\Phi_{eq}$ be the axioms for an equivalence relation. Prove that $\Phi_{eq}$ has continuum many closed complete extensions.
This is a practice question for a prelim, which I have no idea how to start. I think that I may need to use the atomic diagram of a models, but I don't know how to do this.
Any suggestions would be greatly appreciated.
HINT: For every $A\subseteq\Bbb N\setminus\{0\}\cup\{\infty\}$, write a theory stating that for the sizes of the equivalence classes are exactly the members of $A$. Use cardinal arithmetic to prove that there cannot be more than continuum extensions.