Suppose that f is a function from the set a to the set b

Question

Question: Suppose that f is a function from the set A to the set B. Prove that

a) if f is one-to-one, then S_f is a one-to-one function from P(A) to P(B).

b) if f is onto function, then S_f is an onto function from P(A) to P(B).

c) if f is onto function, then S_f -1 is a one-to-one function from P(B) to P(A).

d) if f is one-to-one, then S_ƒ -1 is an onto function from P(B) to P(A).

e) if f is a one-to-one correspondence, then S_ƒ is a oneto-one correspondence from P(A) to P(B) and S_ƒ -1 is a one-to-one correspondence from P(B) to P(A).