Show that there is a one to one correspondence between $\wp(A)$ and the set of all functions $A \to \{0,1\}$

Question

Can I only prove that they have the same cardinality, and, by definition, having the same cardinality implies that there is a bijection between the sets, or do I have to prove that there is an injection and a surjection between the sets?

Answer 1

Let $F$ be the set of all functions from $A \to \{0,1\}$. We can define $\phi: F\to P(A)$ as follows,

$$F(f) = \{ a\in A : f(a) =1\}.$$