The principle of inclusion-exclusion (PIE) is often used to count the number of elements in a union of $n$ sets in terms of an alternating sum of their various intersections:
$$ \left |\bigcup_{i \in [n]} A_i \right| =\sum_{J \subseteq [n]}(-1)^{\left|J\right|-1}\left|\bigcap_{j \in J}A_j\right| $$
where $[n] = \{1,...n\}$. I'm running a psychology experiment that requires subjects to count the number of elements in various sets and in various unions of these sets. I would like to be able to calculate from the behavioral data the $implied$ number of elements in any arbitrary intersection of these sets. It seems this can be performed by writing the PIE in the following recursive form:
$$ \left|\bigcap_{j \in J \subseteq [n]}A_j\right| = \left[\left|\bigcup_{i \in J}A_i\right| - \sum_{S \subsetneq J}(-1)^{\left|S\right|-1}\left|\bigcap_{s \in S}A_s\right|\right](-1)^{\left|J\right|-1} $$
Two questions: (1) is this alternate (recursive) statement of the PIE correct? (2) If so, has anyone seen (or can they provide) a non-recursive solution that expresses the intersection of the $A_j's$ in terms of explicit unions only?