Finding the equivalence class of this relation

Question

I am having this relation:

$$ A=\mathcal P(\mathbb {N} \diagdown 0) , $$

                             A~B :<=>  min A = min B

I haved already proved, that it is a equivalence relation. Now I have to find an equivalence class for this relation. I am knowing the definition of an equivalence class $$[x] := y \in A|\ xRy $$

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What is the amount of the equivalence classes of this relation?

Answer 1

Any subset of $\mathbb{N}$ containing $1$ are equivalent. So the equivalence class for $1$ is $$[1] = \{ \ \{ 1 \} \cup A \ \ | \ \ A \in \mathcal P(\{n \in \mathbb{N} \ | \ n > 1 \}) \ \}$$

Hence in general...

Answer 2

Note that $$\begin{align}f\colon \mathcal P(\mathbb N)\setminus\{\emptyset\}&\to \mathbb N\\S&\mapsto \min S\end{align} $$ is onto and for each $n\in\mathbb N$, the preimage $f^{-1}(n)$ is an equivalence class.