real number and p(N) Equinumerosity

Question

I am not familiar with all rule but I read about infinite set and Cantor's diagonal. I want to prove there is one to one correspondence between power set of natural number and real number. I search a lot but I don't find anything. Is it possible to give me a reference or prove that for me here.
I'm sorry for bad English.
Thanks.

Answer 1

There is a bijection between $P(\mathbb{N})$ And $(0, 1)$. Consider the numbers represented in binary after the point sign. Then if your sub set of naturals contains $n$ you set the $n+1$ didgit to 1 and to 0 otherwise. This gives a unique encoding of each subset in the reals and vise versa.

For example $\{1\}$ is mapped to $0.01$, $\{1,2\}$ is mapped to $0.011$ and $\{3\}$ gives $0.0001$.