Consider the function $f:\Bbb N\to\wp(\Bbb N)$ defined as $f(n) = \emptyset$.
For the question "Find a set $S\subseteq\Bbb N$ s.t. $S\ne D$, but $f(n)\ne S\forall n\in\Bbb N$
how does set $\{137\}$ suffice for an $S$? I don't understand the original function f to begin with. Doesn't $\wp (\Bbb N)$ denote the power set of natural numbers?
Also, would any set containing natural numbers like $\{19\},\{100\}$ work?
$D = \{x \in \mathbb{N} \mid x \notin f(x) \}=\{x \in \mathbb{N} \mid x \notin \emptyset \}=\mathbb{N}.$
You need to find $S\subseteq \mathbb{N}$ such that $S\ne D=\mathbb{N}$ and $\forall n\in \mathbb{N}.\ f(n) = \emptyset\ne S.$
So $S$ must be a non empty proper subset of $\mathbb{N}$. Pick one.