Inclusion_exclusion general formula for intersections?

Question

Assume $A_1,\, A_2, \ldots , A_n$ are subsets of a finite set $S$. Can we find an expression for the size of $S-\{A_1\cap A_2 \cap \ldots \cap A_n\}$ in term of the unions of any number of $A_i$'s (similar to the one we have for $S-\{A_1\cup A_2 \cup \ldots \cup A_n\}$ in term of the intersections of the sets $A_i$'s)

Answer 1

You are talking about the complement of the union which is the same as the intersection of complements.

Thus this follows your formula for the intersection applied to complements.

Answer 2

$|\bigcap_i A_i| = \sum_{i} |A_i| - \sum_{i<j}|A_i\bigcup A_j| + \sum_{i<j<k }| A_i\bigcup A_j\bigcup A_k| - \dots$

every element that belongs to all $A_1...A_n$ may be found exactly once in the left intersection. In the right-hand part it is counted multiple times, like :

$$n\ times - \binom n 2 \ times + \binom n 3 \ times =\cdots = 1 \ time $$ because $(1-1)^n = 0$.

Overall, every element in intersection is counted exactly one time so we get the size of the intersection.