74-hw4

# 74-hw4 - sets That is if A,B,C,D are sets and A ∩ B B ∩...

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MATH 74 HOMEWORK 4: DUE MONDAY 2/26 1. How many elements does each of the following sets have? (a) { 1 , 2 , 3 , 4 } (b) {{ 1 , 2 } , { 3 , 4 }} (c) { 1 , { 2 , 3 } , { 4 , { 5 , 6 }}} (e) { n 2 : n N and 1 n 9 } (f) { n 2 - n : n N and 1 n 9 } (g) { n 2 - 10 n + 9 : n N and 1 n 9 } 2. List all elements of the power set of { 1 , 2 , 3 } . Problems 3-5 In each of the following, when it is possible to give a counterexample, it is possible to give a counterexample where each of A,B,C is a subset of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . If you ﬁnd yourself giving a counterexample, give a counterexample of this form. The main reason I ask you to do this is to develop your ability to look for and exhibit simple counterexamples to false statements. 3. Suppose that A , B , and C are sets, and that A B and that B C . Does it follow that A C ? Prove or give a counterexample. 4(a). Suppose that A , B , and C are sets, and that A B is nonempty, and that B C is nonempty. Does it follow that A C is nonempty? Prove or give a counterexample. 4(b). Depending on your answer to 4(a), do one of the following. If the statement from 4(a) is true, does the statement generalize to four
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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 ﬁnite 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 ﬁnd 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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