Discrete Mathematics with Graph Theory (3rd Edition) 168

Discrete Mathematics with Graph Theory (3rd Edition) 168 -...

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166 Solutions to Exercises (b) We are given lEI = 21MI, IE EB MI = 53, I(E U M)CI = 15. Hence, IE U MI = 97 - 15 = 82. So 82 = lEI + IMI-IEn MI· But 53 = lEI + IMI - 21E n MI (by Proposition 6. 1.1 (e)). Subtracting, we obtain 29 = IE n MI; 29 students are taking both English and Math. Substituting in (*), we have 82 = lEI + IMI- 29 = 31MI- 29, so 31MI = 111, IMI = 37 and lEI = 74. 7. [BB] Let D be the set of delegates who voted to decrease the deficit, E be the set of delegates who voted in favor of the motion concerning environmental issues, and T be the set of delegates who voted in favor of not increasing taxes. (a) This part asks for I(D U E U T)CI. Since ID U E U TI = 1542 + 569 + 1197 - 327 - 92 - 839 + 50 = 2100, we have I(D U E U T)CI = 2300 - 2100 = 200. (b) The set of delegates who voted against increasing taxes but who did not vote in favor of either of the other motions is T" (D U E). Now IT" (D U E)I = ITI-IT n (D U E)I = ITI-I(T n D) U
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Unformatted text preview: (T n E)I = ITI-IT n DI-IT n EI + I(T n D) n (T n E)I = ITI-ITnDI-ITnEI + ITnDnEI = 1197 - 839 - 92 + 50 = 316 8. Let A be the people who dislike the NDP, B the people who dislike the Liberals, C the people who dislike the Conservatives and D the people who dislike the Green Party. We are asked for I (A U B U C U D)CI. We have IA U B U C U DI = IAI + IBI + 101 + IDI -IA n BI -IA n 01 -IA n DI -IB n CI -IB n DI -IC n DI + IA n B n 01 + IA n B n DI + IA n C n DI + IB n C n DI -IAnBnCnDI = 7 + 10 + 11 + 6 - 5 - 5 - 3 - 6 - 4 - 5 + 3 + 2 + 3 + 4 - 2 = 16 and so I(A U B U C U D)CI = IUI-IA U B U C U DI (by Proposition 6.1.1) = 3. 9. Let 0, A, G, and C denote the sets of people who bought orange juice, apple juice, grapefruit juice, and citrus punch, respectively. (a) [BB] We want 10 nAn G n CI. Now 10 U AuG U 01 = 101 + IAI + IGI + 101 -10 n AI -10 n GI - 10 n 01 -IA n GI -IA n CI - IG n 01 + 10 nAn GI + 10 nAn CI + 10nGnCl + IAnGnCl-lonAnGnCl...
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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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