sol4 - MACM 101 — Discrete Mathematics I Outline...

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MACM 101 — Discrete Mathematics I Outline Solutions to Exercises on Induction and Combinatorics 1. Prove that for every positive integer n 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + . . . + n · 2 n = ( n - 1)2 n +1 + 2 . We use induction. Let P ( n ) denote this equality for the integer n . Basis case. P (1) means the equality 1 · 2 1 = (1 - 1)2 n +1 + 2 , which is obviously true. Inductive step. Suppose that P ( k ) is true, that is, 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + . . . + k · 2 k = ( k - 1)2 k +1 + 2 . We have to prove P ( k + 1) : 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + . . . + k · 2 k + ( k + 1) · 2 k +1 = k 2 k +2 + 2 . We have 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + . . . + k · 2 k + ( k + 1) · 2 k +1 = ( k - 1) · 2 k +1 + 2 + ( k + 1) · 2 k +1 = 2 k · 2 k +1 + 2 = k · 2 k +2 + 2 2. Suppose there are n people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. The gossip problem asks for G ( n ) , the minimum number of telephone calls that are needed for all n people to learn about all the scandals. Prove that G ( n ) 2 n - 3 . Denote by P ( n ) the cstatement that G ( n ) 2 n - 3 . Basis step. We prove P (2) . If there are 2 people, then clearly they need only one phone call to exchange all the gossips. Thus G (2) = 1 and 2 · 2 - 3 = 1 , hence, G (2) 2 · 2 - 3 . Inductive step. Suppose that P ( k ) is true, that is, a collection of k people needs at most 2 k - 3 phone calls to share all their gossips. The protocol for k + 1 people goes as follows. The person number k + 1 makes a phone call to a person, say number k , and thus this k th person now know the gossip of the person number k + 1 . Then the first k people make at most 2 k - 3 calls and share all their gossips including the gossip from the person number k + 1 . After that everyone except for ( k + 1) th person knows all the gossips. Now the person number
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This note was uploaded on 10/09/2009 for the course MATH macm 101 taught by Professor Jcliu during the Spring '09 term at Simon Fraser.

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sol4 - MACM 101 — Discrete Mathematics I Outline...

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