the set of all of infinite N subsets

Question

I was thinking about this one for a while and can't find a proper function.

let $M=${$A\in P(N)$|A And A' are infinite}

(for example: the set of all even numbers will be in M, the complement of A will be all odd numbers (of N). )

I need to find the cardinal number of M.

so I know that $M\subseteq P(N)\rightarrow|M|\leq|P(N)=C$

now for the other way, what one to one function can I show to M?

Answer 1

For each subset $A \subseteq P(\Bbb N)$, define $B$ by taking three times each element of $A$, union the set of naturals $\equiv 1 \pmod 3$. Both $B$ and $B'$ are infinite, so we have an injection of $A$ into $M$.

Answer 2

HINT: $\mathcal P(\Bbb N)\setminus M$ is the union of two countable sets.