hw208sol - CS 1050 B: Construction Proofs January 31, 2008...

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CS 1050 B: Construction Proofs January 31, 2008 Solutions to Homework 2 Lecturer: Sasha Boldyreva Problem 2.1, 5 points. Use mathematical induction to prove that 2 n + 3 2 n for all n 4. Base case. n=4: 11 16 - True. Inductive step. Inductive hypothesis: for all n 4 , 2 n + 3 2 n . To prove: for all n 4 , 2( n + 1) + 3 2 n +1 . 2( n + 1) + 3 = (2 n + 3) + 2 2 n + 2 2 · 2 n = 2 n +1 The first inequality used the inductive hypothesis, the second used the fact that 2 2 n for n 1. 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. Base case. n=1. One lined divides the plane into 2 regions. Inductive step. Inductive hypothesis: For n > 0 n lines divide the plane into 2n regions. To show: For n > 0 n+1 lines divide the plane into 2(n+1) regions. Look at n lines first. By the inductive hypothesis, they divide the plane into 2 n regions. Whenever an extra line added, it divides two existing
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hw208sol - CS 1050 B: Construction Proofs January 31, 2008...

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