Proving $|P(\mathbb{N})| $ is equinumerous to these 2 sets.

Question

1. $ \left\{A\in \:P\left(\mathbb{N}\right):\:\:\left|A\right|=\left|\mathbb{N}\right|\right\} $
2. All equivalence,homogenous relations over $ \mathbb{N} $
   For the first set I managed to prove 1 sided function because $ A\subseteq P\left(\mathbb{N}\right)\:\Rightarrow \:\left|A\right|\le P\left(\mathbb{N}\right) $
   But I can't seem to think of another function from $ f:P\left(\mathbb{N}\right)\:\longrightarrow A $ that is a one to one function to use the Cantor-Bernstein theorem.
   For the 2nd set I don't even even have a clue how to make those functions, having trouble defining the equivalence relations.
   Would appreciate any guidance, I translated the assignment from Hebrew so I apologize if it's not clear.

Answer 1

For the first one you could use the fact that if $S$ is uncountable, and $C$ is a countable subset of $S$, then $|S\setminus C|=|S|$ (if you’ve proved it): $\Bbb N$ has only countably many finite subsets. If you want to use the Cantor-Schröder-Bernstein theorem, however, you can get the other injection as follows. Let $E=\{2n:n\in\Bbb N\}$, $O=\Bbb N\setminus E$, and

$$\varphi:\Bbb N\to E:n\mapsto 2n\,,$$

and take

$$h:\wp(\Bbb N)\to\{A\in\wp(\Bbb N):|A|=|\Bbb N|\}:S\mapsto\varphi(S)\cup O$$

as your injection.

For the second question I am assuming that a homogeneous equivalence relation is one for which all of the equivalence classes have the same cardinality. Recall that there is a bijection between equivalence relations on $\Bbb N$ and partitions of $\Bbb N$, so we’re looking at the set of all partitions of $\Bbb N$ whose parts are all of the same cardinality.

First note that $|\{A\in\wp(E):|A|=|\Bbb N|\}|=|\wp(E)|=|\wp(\Bbb N)|$. Then observe that for each $A\in\wp(E)$ such that $|A|=|\Bbb N|$, $\{A,\Bbb N\setminus A\}$ is a homogeneous partition of $\Bbb N$. Thus, there are at least $|\wp(\Bbb N)|$ homogeneous partitions of $\Bbb N$: this indirectly gives you an injection from $\wp(\Bbb N)$ to the set of homogeneous partitions of $\Bbb N$.

For the other inequality note that if $\mathscr{P}$ is a partition of $\Bbb N$, we can list the parts of $\mathscr{P}$ in increasing order of their smallest elements. This gives us a sequence $\langle P_k:k\in\Bbb N\rangle$ of subsets of $\Bbb N$: if $|\mathscr{P}|=n$, we let $P_k=\varnothing$ for $k\ge n$. (Note that my $\Bbb N$ includes $0$.) We now have an injection from the set of homogeneous partitions of $\Bbb N$ to $\wp(\Bbb N)^{\Bbb N}$. I expect that you already know that $|\wp(\Bbb N)|=\left|\{0,1\}^{\Bbb N}\right|$, so

$$\left|\wp(\Bbb N)^{\Bbb N}\right|=\left|\left(\{0,1\}^{\Bbb N}\right)^{\Bbb N}\right|=\left|\{0,1\}^{\Bbb N\times\Bbb N}\right|=\left|\{0,1\}^{\Bbb N}\right|=|\wp(\Bbb N)|\,.$$