I'm afraid this may be an embarrassing question but I've never chanced upon what a set - valued function is hitherto.
Let $S = \Pi_{i=1}^{N} \Delta(S_{i})$. Let $F:S \rightarrow 2^{S}$
The explicit form for $S$ is $S = (\Delta(S_{1}, \cdot \cdot \cdot ,\Delta(S_{N}))$. What is $2^{S}$ explicitly?
This is an alternative notation to $\mathcal P(S)$, power set of $S$. It's the set of every subset of $S$.
The notation is such because the cardinality of $\mathcal P(S)$ is $2$ to the power of the cardinality of $S$:
$$\vert \mathcal P(S) \vert = \vert 2^S \vert = 2^{\vert S \vert}$$
Another reason, perhaps more abstract:
The set of all functions with domain $X$ and codomain $Y$ is denoted $Y^X$, (because $\vert Y^X \vert = \vert Y \vert^{\vert X \vert}$). If you think of $2$ as the set $\{0,1\}$ then:
$$2^{S} = \{0,1\}^S = \left\{ { f\vert f :S \to \{0,1\}} \right\}$$
Is the set of all functions sending elements of $S$ to $0$ or $1$. You can think of elements of $S$ being sent to a subset if it's mapped to $1$ or omitted from a subset if it's mapped to $0$. Thus the set all combinations of either including or omitting the elements of $S$ gives you all possible subsets.