I am wondering if for any given $x \in P(\Bbb{N})- \{\emptyset\}$ we can find an equivalence relation such that it will have an equivalence class equal to $x$.
Extend of this question is whether for set $R$ of all relations in $\Bbb{N}$, the following applies: $$\bigcup_{r \in R} \Bbb{N}/_r = P(\Bbb{N}) - \{\emptyset\}$$
If you have a subset $A$ of $\mathbb{N}$, define an equivalence relation $n \sim m$ if and only if $m,n \in A$ or $m,n \not \in A$. It's not hard to show that this is an equivalence relation, and that $A$ is an equivalence class of that relation.