Discrete Mathematics with Graph Theory (3rd Edition) 184

Discrete Mathematics with Graph Theory (3rd Edition) 184 -...

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182 Solutions to Review Exercises 6. Let 8 be the set of cars with stereo systems, A the set of cars with air conditioners and R the set with sun roofs. We are given that 181 = 3D, IAI = 3D, IRI = 40, 1(8 n A) U (8 n R) U (A n R) I = 30 and 18 nAn RI = 10. By the Principle of Inclusion-Exclusion, 30 = 1(8 n A) U (8 n R) U (A n R)I = 18 n AI + 18 n RI + IA n RI -1(8 n A) n (8 n R)I -1(8 n A) n (A n R)I -1(8 n R) n (A n R)I + 1(8 n A) n (8 n R) n (A n R)I = 18nAI + 18nRI + IAnRI-218nAnRI· Thus, 18 n AI + 18 n RI + IA n RI = 30 + 2(10) = 50. It follows that the number of car with at least one feature is 18 U A U RI = 181 + IAI + IRI - 18 n AI - 18 n RI - IA n RI + 18 nAn RI = 30 + 30 + 40 - 50 + 10 = 60. Since the number of cars with exactly one feature is the number with at least one feature less the number with at least two, the number with exactly one is 60 - 30 = 30. 7. If n objects are put into m boxes and n > m, then some box must contain at least r {;, 1 objects.
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