Concrete Mathematics - Recurrent Problems - Warmups - (5)
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Chapter One: Recurrent Problems
Warmups
5. A "Venn diagram" with three overlapping circles is often used to illustrate the eight possible subsets associated with three given sets. Can the sixteen possibilities that arise with four given sets be illustrated by four overlapping circles?
Well, this is also a recurrent problem. How many possible regions can we have at most by four overlapping circles? The answer is 14. Because when three circles overlap, it can make 8 regions at most. The three existing circles can cut the fourth circle into 6 arcs by overlapping at most, for each circle can cut a circle into 2 arcs at most. And each arc in the fourth circle can cut an existing region into two regions, so there would be 6 new regions by overlapping the fourth circle at most. Thus there would be 14 regions at most by four overlapping circles. But four given sets can generate 16 different sets, so there would be not enough regions such that one region corresponds to one set.
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