Cardinal number of a set

Question

What is the cardinal number of the following set and how do you prove it? $$ \left \{ f:\mathbb{N}\rightarrow \left \{ 0,1 \right \} :\left|f^{-1}[\left \{ 1\right \}]\right|=\left|f^{-1}[{\left \{ 0 \right \}}]\right| \right \} $$

Answer 1

Its cardinality is $2^{\aleph_0}$. For example map elements $3n$ to $0$ and $3n+1$ to $1$. Then for elements, $3n+2$, divide them into two classes, there are $2^{\aleph_0}$ many ways to do this, and map one class to $0$, the other to $1$.