hw1 - Introductory Combinatorics by Brualdi

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Prepared by Maya Ahmed April 3, 2003 1. Brualdi 2.4 Show that if n + 1 integers are chosen from the set { 1 , 2 , 3 , .., 2 n } , then there are always two which differ by 1. Answer: Partition the set { 1 , 2 , 3 , .., 2 n } into n subsets as following { 1 , 2 }{ 3 , 4 } , .... , { 2 n - 1 , 2 n } . Now if we choose n + 1 distinct numbers from { 1 , 2 , 3 , .., 2 n } , then two must come from the same subset by pigeonhole princi- ple. These two numbers must then differ only by 1. 2. Brualdi 2.5 Show that if n + 1 integers are chosen from the set { 1 , 2 , 3 , .., 3 n } , then there are always two which differ by at most two. Answer: Partition the set { 1 , 2 , 3 , .., 3 n } into n subsets as following { 1 , 2 , 3 } , { 4 , 5 , 6 } , .... , { 3 n - 2 , 3 n - 1 , 3 n } . Now if we choose n + 1 distinct numbers from { 1 , 2 , 3 , .., 3 n } , then two must come from the same subset by pigeonhole principle. These two numbers must then differ by no more than two. 3. Brualdi 2.8 Use the pigeonhole principle to prove that the decimal expansion expansion of a rational number m/n eventually is repeating. Answer: Assume m < n , since if m > n then after dividing m by n we get m = pn + r and the proof will apply to r/n . Let
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hw1 - Introductory Combinatorics by Brualdi

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