Notation for a function with multiple return values

Question

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 $V$.

Answer 1

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 $f:V\to \mathcal{P}(C)$.

Answer 2

Let $S = 2^C$, now $f:V \to S$.