Finding equivalence classes of an equivalence relation on $\mathcal{P(\mathbb{Z})}$

Question

We define a relation R on the set $\mathcal{P}(\mathbb{Z})$ where we let $S$ and $T$ be sets for which $S,T\in{\mathcal{P}(\mathbb{Z})}$ and define S to be related to T if and only if there exists a bijection from S to T.

\begin{equation} R = \bigg\{(S,T)\in\mathcal{P}(\mathbb{Z})\times \mathcal{P}(\mathbb{Z}): S\underset{\text{bijection}}{{\longrightarrow}}T\bigg\} \end{equation}

The correspondig equivalence classes of $\{0\}$ and $P = \{2,3,5,7,...\}$ are, \begin{align*} \left[\{0\}\right] &= \bigg\{S\in\mathcal{P(\mathbb{Z})}: S\underset{\text{bijection}}{\longrightarrow}\{0\}\bigg\} = \bigg\{\{n\}:n\in\mathbb{Z}\bigg\}\\ \\ \left[P\right] &= \bigg\{S\in\mathcal{P(\mathbb{Z})}: S\underset{\text{bijection}}{\longrightarrow}P\bigg\} = \end{align*} Now for the equivalence classes for $\{0\}$ we just have a set which contains the sets of just all individual elements in $\mathbb{Z}$ which create bijections with the set $\{0\}$. Now for the primes we have $\mathbb{N}$ and $\mathbb{Z}$ itself for example since they all have same cardinality of $\aleph_0$. But now there are sets like $\{2,4,6,8,10,...\}$ and $\{1,3,5,7,9,...\}$ and ... there are an infinite number of elements in $\mathcal{P(\mathbb{Z})}$ which form a bijection with the primes or have same cardinality if you wish. Is this correct? For the primes what should I include all such sets should I just write that the equivalence relation of the primes is the set of sets which have same cardinality as $\aleph_0$?

Answer 1

To explicitly answer two of your questions in the comments:

• "would I say that the equivalence class of the primes is just the set of all infinite subsets of $\mathbb Z$?": Yes, that would be correct, although it is also the equivalence class of $\mathbb Z$ itself, and I personally wouldn't bother to involve the set of primes. I think it complicates things for no benefit.

• "How should I explicitly go about writing out the equivalence class for the primes ...?": You shouldn't. In fact it's most likely that you can't (although proving that something can't be done requires us to get very clear about just what 'explicitly writing' means, and let's not go there).

I'm wondering if you're coming to math from a computer programming background. In programming we're expected to write algorithms that generate things. In math you can say "It's the set of all subsets with exactly two elements", and be done.