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Unformatted text preview: n . 3. Dene a set M Z 2 as follows (1) (3 , 2) M (2) If ( x,y ) M , then (3 x-2 y,x ) M Use structural induction to prove that elements of M always have the form (2 k +1 + 1 , 2 k + 1), where k is a natural number. (The point of this problem is to learn how to use structural induction, so you may not rephrase this into a normal proof by induction on k .) 4. The Fibonacci trees T k are a special sort of binary trees that are dened as follows. Base: T 1 and T 2 are binary trees with only a single vertex. Induction: For any n 3, T n consists of a root node with T n-1 as its left subtree and T n-2 as its right subtree. Use structural induction to prove that the height of T n is n-2, for any n 2. (Again, use structural induction rather than looking for an explicit induction variable n .) 1...
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- Spring '10