hw1sol - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 1 Due at the beginning of class on Wed, September 26, 2007 Problem 1 Prove that if x ≥ - 1, then for any integer n 0, (1 + x ) n > nx . Solution: We will strengthen the inductive hypothesis: we will prove that for any integer n 0, (1+ x ) n nx +1. The problem statement follows directly. (Note: induction on the problem statement directly does not work) Base case: n = 0, then (1 + x ) n = (1 + x ) 0 = 1 = 0 x + 1 = nx + 1 — OK Inductive hypothesis: Suppose for some integer n = n 0 0, (1 + x ) n nx + 1. Inductive step: Consider n = n 0 + 1. Using the inductive hypothesis (1+ x ) n = (1+ x ) n 0 (1+ x ) ( n 0 x +1)(1+ x ) = n 0 x + n 0 x 2 +1+ x ( n 0 +1) x +1 = nx +1 and we are done. Problem 2 Show that at a party of n people, there are two people who have the same number of friends in the party. (Friendship is symmetric) Solution: Bin the people at the party by the number of friends each person has. Each person can be friends with between 0 and n - 1 people (inclusively). This is n bins and does not yet solve the problem. However, since friendship is symmetric, if a person has 0 friends, there is no person with n - 1 friends (who would be friends with everyone). So, there are at most n - 1 non-empty bins, with n people in them. By the pigeonhole principle, there is some bin that contains at least two people, and the people in that bin have the same number of friends in the party. Problem 3 There are two children sitting on a (very long) bench. The child on the left is a boy, the child on the right is a girl. Every minute, either two children arrive and sit down next to each other on the bench (possibly squeezing between two children who are already sitting), or two children who had been sitting next to each other get up off the bench and leave.
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