let x be a set and P(x) is one of the power sets of x,
show that there is no onto function which is from x to P(x).
Suppose that $\displaystyle f:X \mapsto P(X)$ is any mapping from a set to its power set.
Now define $\displaystyle A = \left\{ {y \in X:y \notin f(y)} \right\}$. Clearly $\displaystyle A \in P(X)$.
If $\displaystyle f$ were a surjection then $\displaystyle \left( {\exists a \in X} \right)\left[ {f(a) = A} \right]$.
Work with until you see the contradiction