For any set $B$ let $\mathcal{P}(B)$ denote the set of all subsets of $B$.
Let $A$ be an infinite set and suppose there exists a surjection $f : A \mapsto \mathcal{P}(A)\setminus A$. Consider the set $D := \{a : a \in A\ \textrm{but}\ a \notin f(a)\}$. Is it ever possible that the set $A\setminus D$ is empty?