How can we construct a bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$?
Here is what I know:
$\mathcal{P}(\mathbb{N}) = \{A | A \text{ is a subset of } \mathbb{N}\}$
Both $\mathcal{P}(\mathbb{N})$ and $(0,1)$ are uncountable, so such bijection exists <--- incorrect
There are uncountable many functions from $(0,1)$ to $\mathbb{N}$
So I seek a bijection function that takes a set $A \in \mathcal{P}(\mathbb{N}) \mapsto x \in (0,1)$ and back...I don't see how this can be done
Is anyone familiar with this result?