# hw208 - Proof Let P n be the predicate “n = n 1” To...

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CS 1050 B: Construction Proofs January 24, 2008 Homework 2 Lecturer: Sasha Boldyreva Due: January 31, 2008 Assignment 2.01 Do the assigned reading. Assignment 2.02 Indicate how much time did you spend on this homework. Problem 2.1, 5 points. Use mathematical induction to prove that 2 n + 3 2 n for all n 4. Problem 2.2, 5 points. Use mathematical induction to show that n distinct lines in the plane passing through the same point divide the plane into 2 n regions. Problem 2.3, 8 points. Let a 1 = 2 , a 2 = 9, and a n = 2 a n - 1 + 3 a n - 2 for n 3. Use strong induction (the Second Principle of Mathematical Induction) to show that a n 3 n for all positive integers n . Problem 2.4, 5 points. Find the error in the following proof of this “theorem”: Theorem: Every positive integer equals the next largest positive integer.
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Unformatted text preview: Proof: Let P ( n ) be the predicate “n = n + 1”. To show that P ( k ) → P ( k + 1), assume that P ( k ) is true for some k , so that k = k + 1. Add 1 to both sides of this equation to obtain k + 1 = k + 2, which is P ( k + 1). Therefore P ( k ) → P ( k + 1) is true. Hence P ( n ) is true for all positive integers n . Problem 2.5, 6 points. Sharing a chocolate bar. Problem 10 from Section 4.2 of Rosen’s textbook. Problem 2.6, 6 points. Describe a recursive algorithm for computing 5 2 n where n is a nonnegative integer. Problem 2.7, 6 points. Problem 38 from Section 4.4 of Rosen’s textbook....
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