Let A be a set , prove that $|\{f|f:A\rightarrow \{0,1\}\}|=|P(A)|$
I tried to prove that $g:|\{f|f:A\rightarrow \{0,1\}\}|\rightarrow |P(A)|$ is one to one and onto
but I didn't find the right function and I don't know how to find cardinality of functions
thoughts?
Hint: define $g: \{f|f:A\rightarrow \{0,1\}\} \rightarrow P(A)$ by $$g(f) = f^{-1}(\{1\})$$ where the RHS is the inverse image of $\{1\}$, namely all the elements in $A$ that are mapped to $1$ (you could equally well take the inverse image of $\{0\}$, the choice is arbitrary). Now show that $g$ is one-to-one and onto.
