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To generalize some formula to finit sets union what is the best way to start?

let me give an example: $$n(A\cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C + n (A \cap B \cap C) $$

How can I generalize some formula to $n$ elements, like $n(A_1\cup A_2 \cup A_3 \cup A_4 \cdots \cup A_k)$?

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Check wikipedia for inclusion-exclusion theorem – Alex Sep 22 '13 at 18:45
up vote 1 down vote accepted

This is known as Inclusion-Exclusion, and it can be formulated as $$ \left| \cup_{i \in \{1,2,\ldots,n\}} A_i \right|=\sum_{S \subseteq \{1,2,\ldots,n\}} (-1)^{|S|-1} \left| \cap_{j \in S} A_j \right|.$$ There are proofs of it at the Wikipedia site.

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