Chapter 2, Problem 16SE

*16. Suppose that f is a function from the set A to the set B. Prove that

if f is one-to-one. then $S_f$ is a one-to-one function from $P\left(A\right)$ to $P\left(B\right)$.

if f is onto function, then $S_f$ is an onto function from $P\left(A\right)$ to $P\left(B\right)$.

if f is onto function, then $S_{f-1}$ is a one-to-one function from $P\left(B\right)$ to $P\left(A\right)$.

if f is one-to-one, then $S_{f-1}$ is an onto function from $P\left(B\right)$ to $P\left(A\right)$.

if f is a one-to-one correspondence, then $S_f$ is a one-to-one correspondence from $P\left(A\right)$ to $P\left(B\right)$ and $S_{f-1}$ is a one-to-one correspondence from $P\left(B\right)$ to $P\left(A\right)$. [Hint: Use parts (a)−(d).]