A1 - MATH 239 ASSIGNMENT 1 Due Friday, 14th May at 11:45...

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MATH 239 ASSIGNMENT 1 Due Friday, 14th May at 11:45 a.m. 1. Let S n denote the set of sequences of 0’s and 1’s of length n , written as vectors. For example, S 2 = { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) } . Give a combinatorial proof that, for all n 1, the number of sequences in S n having an odd number of 1’s is equal to the number of subsets of { 1 ,...,n } with an odd number of elements. 2. Let n be a positive integer. If T is a subset of { 1 ,...,n } , we say that T is an n -palindrome if, for every i ∈ { 1 ,...,n } , if i T then n + 1 - i T . For instance, { 1 , 2 , 4 , 5 } is a 5-palindrome, but is not a 6-palindrome. Examples of 6-palindromes include { 2 , 5 } and the empty set. Show with a combinatorial proof that the number of n -palindromes equals 2 n/ 2 for n even, and 2 ( n +1) / 2 for n odd. (Hint: A set of cardinality k has 2 k subsets.) 3. For i 1 let U i be the set of ordered pairs of positive integers such that the larger one is equal to i . For example,
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