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Homework 9

# Homework 9 - n as an element Let B n denote the set of...

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MATH 74 HOMEWORK 9 (DUE WEDNESDAY NOVEMBER 7) Exercises 1,2. Let A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . 1. Let X denote the set of subsets of A that have an odd number of elements. Let Y denote the set of subsets of A that have an even number of elements. Give an example of a bijection f : X Y (and prove that it is a bijection). 2. Let S denote the set of 3-element subsets of A , and let T denote the set of 7-element subsets of A . Give an example of a bijection g : S T (and prove that it is a bijection). Exercises 3,4,5. In these exercises let N n = { 1 , 2 , . . . , n } , and let K n denote the collection of all subsets of N n that contain no consecutive pair of numbers. Thus for example { 1 , 3 , 5 } ∈ K 5 and { 2 , 4 } ∈ K 5 but { 1 , 3 , 4 } 6∈ K 5 because { 1 , 3 , 4 } contains the consecutive pair 3 , 4. 3. Write down— that is, list the elements of— the sets K 1 , K 2 , and K 3 . (Don’t forget about the empty set). 4. For n = 1 , 2 , . . . let A n denote the set of elements of
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Unformatted text preview: n as an element. Let B n denote the set of elements of K n that do not contain n as an element. Notice that A n and B n are disjoint sets and that K n = A n ∪ B n . 4(a). Write down the sets A 3 and B 3 . 4(b). Assume n > 2. Write down a rule that gives a bijection between A n and K n-2 . (You do not need to prove it is a bijection, just specify the rule.) 4(c). Assume n > 2. Write down a rule that gives a bijection between B n and K n-1 . (Hint: don’t try too hard.) 5. For n = 1 , 2 ,... , let k n = #( K n ). 5(a). Prove that if n > 2, k n = k n-1 + k n-2 . 5(b). How many subsets of the set A in Exercise 1 contain no consecutive pair of numbers? [You may assume the result of 5(a) if you did not prove it.]...
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