Let X,Y be subsets of the universal set S. Suppose f:X to Y is a function. Define the lift of f to 2^X , F:2^X to 2^Y by F(A)=f(A), A in 2^X. Show the following:

a) F is one-to-one if and only if f is one-to-one.

b)F is onto if and only if f is onto.

I know that 2^X is the power set of X, meaning that 2^X is the set of all subsets of X. Similarly, 2^Y is the set of all subsets of X. But I have no experience with lifts at all. To prove a), I will first assume that F is one-to-one. Then for a1,a2 in 2^X F(a1)=F(a2) implies a1=a2. But I don't understand how the lift is relating to f, so I cannot complete the proof. Thank you for your help.