How to define an explicit bijection from P(N) to 2^N

Question

How do I define an explicit bijection between the power set of N and $2^N$ with $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?

Answer 1

Assign to every subset $S \subseteq \mathbb{N}$, the map $$ \begin{array}{rcl} \chi_S \colon \mathbb{N} &\to &\{0,1\}\\ x & \mapsto & \left\{ \begin{array}{l} 1 \quad\text{ if } x\in S, \\ 0 \quad\text{ if } x \notin S. \end{array} \right. \end{array} $$