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If A and B are two different nonempty sets, how many distinct sets can be formed with these sets using as many unions,intersections,complements and parentheses as desired.

Four sets are fundamental:$A$,$B$,$A \cup B$,$A \cap B$.

Other sets are A \cup B',$A' \cup B$,$A \cap B'$,$A' \cap B$,$A' \cup B'$,$A' \cap B'$. Any other sets are possible.

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    To get a clear answer, you have to assume that A and B are not just distinct but also "as independent as possible" in some sense. Otherwise, it could be the case that A = {1} and B = {1,2}, and you only get 8 possible sets.2011-01-17

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$16$. The corresponding Venn diagram has four parts, and you can get any combination of those four parts.

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    @Vinod $2^{2^n - 1}$ to be accurate.2012-04-08