I apologize in advance for my phrasing, I'm translating this from another language.
let R be a relation under P(N) where ∀A,B ∈P(N), (A,B)∈R if and only if |A|=|B|.
R is an equivalence relation, therefore it splits P(N) to equivalence classes.
The question is what's the cardinality of the equivalence class that N is a member of. I have a feeling it's c (|R|), but I don't know how to show it.