# Less restrictive set intersection

I have a question that may be trivial but I just can't find an appropriate answer on the Internet. The inclusion-exclusion principle can be used to discern the cardinality of the union among sets $\bar{a} =\lvert \cup_{i=1}^n A_i \rvert$. Similarly, I can use it to count intersections, $\bar{b} =\lvert\cap_{i=1}^n A_i \rvert$, in which case $\bar b$ is the number of elements that belong to all sets. Now, I would like to count the number of elements that are part of any intersection between the sets $A_i$ taken by two, i.e., $\bar c = \lvert \cup \left\{ A_i \cap A_j \right\} \rvert$. Does this have a particular name? I guess it is a less restrictive intersection operation, as $\bar c \geq \bar b$.

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It's not clear what the question is. Are you looking for a way to compute $\bar{c}$? In that case the inclusion-exclusion principle would seem to do the job, and you'll likely end up with a subexpression of the result of I-E applied to $|\bigcup_{i=1}^n A_i|$. – Paul McKenney Feb 19 '13 at 15:54
Also, it seems that the set-theory tag is a bit inappropriate; combinatorics is probably better. – Paul McKenney Feb 19 '13 at 15:55
Given sets $A_1, \dots, A_n$, I would like to find the set of elements that belong to any intersection of sets taken by two, i.e., $\bigcup_{i \neq j} A_i \cap A_j$. Is there a name for this? – Alejandro Marcos Aragon Feb 19 '13 at 16:35

Take a look at Wilf's "generatingfunctionology", he derives a marvelous formula that gives the answer to such questions very simply.

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Hi vonbrand, thanks for your answer. Got the book, but I can't seem to find such magical equation. Could you please point out which one you're talking about? – Alejandro Marcos Aragon Feb 19 '13 at 23:24
Section 4.2 "A generatingfunctiological view of the sieve method" – vonbrand Feb 19 '13 at 23:33