Discrete Mathematics with Graph Theory (3rd Edition) 183

Discrete Mathematics with Graph Theory (3rd Edition) 183 -...

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Chapter 6 181 Chapter 6 Review 1. By the Principle ofInclusion-Exclusion, IA n BI = IAI + IBI- IA UBI, so IA n BI > O. 2. Let A, B, C, D be the sets of integers between 1 and 2000 (inclusive) which are divisible by 2, by 3, by 5, and by 7, respectively. We want I(A U B U C U D)I. Now IA U B U C U DI = IAI + IBI + ICI + IDI We have - IA n BI - IA n CI - IA n DI - IB n CI - IB n DI - IC n DI + IA n B n CI + IA n B n DI + IA n C n DI + IB n C n DI -IAnBnCnDI· IAI = l 20 2 00 J = 1000, IBI = l20 3 00 J = 666, ICI = l 20 J = 400, IDI = l20~O J = 285, IA n BI = l20 6 00 J = 333, IA n CI = l2~gO J = 200, IB n CI = l2~~O J = 133, IB n DI = l2~~O J = 95, IA n DI = l2~~O J = 142, IC n DI = l2~~O J = 57, IA n B n CI = l2~gO J = 66, IA n B n DI = l2~~O J = 47, IA n C n DI = l2~gO J = 28, IB n C n DI = l2 I 0 0 0 5 0 J = 19, IA n B n C n DI = l2 2 0 I O O O J = 9. SO IA U B U C U DI = 1542. 3. (a) A palindrome is a word which is spelt the same way forwards as backwards, such as SUNUNUS. (b) We subtract from the total number of seven-letter palindromes which begin with S the number
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