How can Russell’s paradox help me in this problem?

Question

if set $A$ is finite, then $|| < 2^{||} = |()|$, and so there is no surjection from set $A$ to its powerset.show that this is still true if $A$ is infinite. Hint: Remember Russell’s paradox and consider $\{ \in \mid \notin ()\}$ where $f$ is such surjection.

Answer 1

Suppose that $f:A \rightarrow \mathcal P (A)$ is a surjection. Then for any set $B \in \mathcal P(A)$, there exists some $x \in A$ such that $f(x)=B$ (that is what surjection means). Now, letting $B$ equal that set in the hint, what can you say?