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Mohamed Gouda							CS 386 S
Spring 2008							Quiz#2
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Let S be a set of d elements (d >= 4). Also, let C be a collection of k subsets
of S such that all 3 distinct subsets B, B', B'' in C

     	       B is not a subset of B' Union B''

Show that k <= d.
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(Solution)
We have a counterexample as follows.

S = {1,2,3,4,5,6,7}
C = {(1,2,3), (1,4,5), (1,6,7), (2,4,7), (2,5,6), (3,4,6), (3,5,7)}

|C| = 7

S = {1,2,...,7,1',...,7',1'',...,7''}

|S| = 21

C = {(1,2,3), (1,4,5), (1,6,7), (2,4,7), (2,5,6), (3,4,6), (3,5,7),
(1',2',3'), (1',4',5'), (1',6',7'), (2',4',7'), (2',5',6'), (3',4',6'), 
(3',5',7'), (1'',2'',3''), (1'',4'',5''), (1'',6'',7''), (2'',4'',7''), 
(2'',5'',6''), (3'',4'',6''), (3'',5'',7''),(1',1'',1''')}

|C| = 22