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Unformatted text preview: sets? That is: if A,B,C,D are sets and A B , B C , and C D are nonempty, is A D always nonempty? Prove this or give a counterexample. Is a similar statement true for an arbitrary nite number of sets? If you found a counterexample in 4(a), does your counterexample have the property that both #( A B ) > 1 2 #( A ) and #( B C ) > 1 2 #( B )? (This would be a example where A C is empty, despite the fact that a majority of the elements of A are elements of B , and a majority of the elements of B are elements of C .) If your example did not have this property, can you nd one that does? Give an example, or prove that no such example is possible. 5. Suppose that A , B , and C are sets, and that each of the sets A B , B C , and A C is nonempty. Does it follow that A B C is nonempty? Prove or give a counterexample. 1...
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- Fall '07