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A "Venn diagram" with three overlapping circles is often used to illustrate the eight possible subsets associated with three given sets:

such as the image in http://robertdaylin.files.wordpress.com/2008/10/venn-diagram1.png.

Can the sixteen possibilities that arise with four given sets be illustrated by four overlapping circles?

I have no idea about this problem. Can anyone help me?

I have know the answer.But another problem come, what's the maximum number of regions with n overlapping circles?

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 How many intersection points could you get with four circles? And how many would you need for a Venn diagram. One can achieve this with four (congruent) ellipses. See Ruskey and Weston's survey at combinatorics.org/Surveys/ds5/VennEJC.html . – Robin Chapman Sep 8 2010 at 12:22
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 The book Concrete Mathematics has full answers for most exercises (except those marked as research problems), so it's hardly needed to ask on MO. – rgrig Sep 8 2010 at 12:33

closed as too localized by Robin Chapman, Steve Huntsman, Akhil Mathew, Franz Lemmermeyer, David Speyer Sep 8 2010 at 13:17

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