Cardinality of power set and binary sequence 
Let $A$ be a set and $P(A)$ be the power set of $A$. Define $B(A)$ as
  the set of all functions $F:A\rightarrow\{0,1\}$. For example,
  $B(\mathbb{N})$ is the set of all binary sequences. Prove that $P(A)$
  has the same cardinality as $B(A)$.

When $A$ is finite, this is easy to prove. I am interested in other cases; for instance when $A$ is countably infinite or uncountable. I am also a bit confused with the definition of $B(A)$. Could anyone help me with this one please?

Hint. Consider, for each subset $B$ of $A$, its characteristic function $\chi_B$ defined by
$$
\chi_B(x)= \begin{cases}1&\text{if $x \in B$}\\
0&\text{otherwise}\end{cases}
$$
