Chapter 2.5, Problem 40E

Show that if S is a set, then there does not exist an onto function f from S to $P\left(S\right)$ , the power set of S. Conclude that $\left|A\right|<\left|P\left(S\right)\right|$ . This result is known as Cantor’s theorem. [Hint: Suppose such a function f existed. Let $T=\left\{s\in |s\cancel{\in }f\left(s\right)\right\}$ and show that no element s can exist for which $f\left(s\right)=T$ .]