Prove that these two sets have equal cardinality

Question

Let $\mathscr{F}$ be the set of functions from $X\rightarrow \{0,1\}$. Then card$(\mathscr{P}(X))$=card$\mathscr{F}$.

I am really stuck on this one. How do I show that there is a bijection between those two sets.

Answer 1

Represent each subset $S$ in $\mathscr{P}(x)$ by the function $f:X\rightarrow \{0,1\}$ such that, for $x \in X$, $f(x)=1$ if $x \in S$ and $f(x)=0$ otherwise.