I want to define a function $f$ whose domain is given by the set $V$ whose return value is a subset of $C$.
Please correct me if I am wrong, I assume that $f : V \rightarrow C$ would mean that the function returns a single value from the set $C$. My question is, how do I denote that the function returns a subset of values from $C$ for every value in $V$.
Best Answer
For any set $S$, the notation $\mathcal{P}(S)$ denotes the power set of $S$ (Wikipedia link), whose elements are themselves the subsets of $S$.
Thus, what you are after is a function $f:V\to \mathcal{P}(C)$.