Are there countably or uncountably many infinite subsets of the positive even integers? Let $S$ be the set of all infinite subsets of $\mathbb N$ such that $S$ consists only of even numbers.
Is $S$ countable or uncountable?
I know that set $F$ of all finite subsets of $\mathbb N$ is countable but from that I am not able to deduce that $S$ is uncountable since it looks hard to find a bijection between $S$ and $P(\mathbb N)\setminus F$. Also I am not finding the way at the moment to find any bijection between $S$ and $[0,1]$ to show that $S$ is uncountable nor I can find any bijection between $S$ and $\mathbb N$ or $S$ and $\mathbb Q$ to show that it is countable. So I am thinking is there some clever way to show what is the cardinality of $S$ by avoiding bijectivity arguments?
So can you help me?

The set of even numbers is countably infinite, and so equinumerous to $\Bbb N$ (explicit bijection given by $n\mapsto\frac12n$). With $F$ the set of finite subsets of $\Bbb N$, as you say, we can use the map $f(n)=\frac12n$ to get a bijection $g:S\to P(\Bbb N)\setminus F$, given by $$g(A)=\{f(n):n\in A\}.$$ I leave it to you to prove that it's bijective.
