Discrete Mathematics with Graph Theory (3rd Edition) 172

# Discrete - 170 Solutions to Exercises(a This part asks us for IA U B U C U D I and we have IA U B U C U D I = IAI IBI ICI IDI IA n B I IA n CI IA n

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170 Solutions to Exercises (a) This part asks us for IA U B U C U DI and we have IA U B U C U DI = IAI + IBI + ICI + IDI - 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 = 3333 + 2000 + 1428 + 909 - 666 - 476 - 303 - 285 - 181 - 129 + 95 + 60 + 43 + 25 - 8 = 5845. (b) This part asks for I(A n B) '-. (C U D)I = IA n BI -I(A n B) n (C U D)I· Now (A n B) n (C U D) = (A n B n C) U (A n B n D), so I(A n B) n (C U D)I = IA n B n CI + I(A n B n D)I-I(A n B n C) n (A n B n D)I = IAnBnCl + I(AnBnD)I-I(AnBnCnD)1 = 95 + 60 - 8 = 147. The answer is 666 - 147 = 519. (c) This question asks for the number of elements in ((AnBnC) '-.D) U ((AnBnD) '-.C) U ((AnCnD) '-.B) U ((BnCnD) '-.A). Since these sets are pairwise disjoint, that is, the intersection of two of the four is empty, the number of elements in the union is the sum of the elements in each set. Since I(A n B n CI '-. DI = IA n B n CI-IA n B n C n DI = 95 - 8 =
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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