Discrete Mathematics with Graph Theory (3rd Edition) 194

Discrete Mathematics with Graph Theory (3rd Edition) 194 -...

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192 Solutions to Exercises 28. [BB] A and B are not mutually exclusive: The ordered pair (2, 2) is in each. [BB] A and e are mutually exclusive: (2,2) is not in e because 4 ~ 70. A and D are mutually exclusive because 2 + 2 is not divisible by 5. Band e are not mutually exclusive: The ordered pair (2,5) is in each. Band D are not mutually exclusive: The ordered pair (2,3) is in each. e and D are mutually exclusive: D consists of the ordered pairs (1,4),(2,3),(3,2),(4,1),(4,6),(5,5),(6,4) and none of these is in e. 29. The result follows from the facts that A = (A n B) U (A nBC) and (A n B) n (A nBC) = 0. 30. (a) [BB] P(A) = P(A n B 1) + P(A n B2) + . .. + P(A n Bn). Note that when n = 2, Bl n B2 = 0 and Bl U B2 = S, so B2 = Bf and the identity here indeed generalizes that of Exercise 30. (b) The result follows from the facts that A = (A n B 1 ) U (A n B2) U . .. U (A n Bn), and that (A n B
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Unformatted text preview: i ) n (A n B j) = 0 for i =1= j. 31. (a) [BB] Using Proposition 6.1.1(c), we have P(A B) = IA" BI = IAI -IA n BI = ~ _ IA n BI = P(A) _ P(A B) " lSI lSI lSI lSI n . (b) From (a), P(A" B) = P(A) -P(A n B) = P(A) -(P(A) + P(B) -P(A U B)) = P(A U B) -P(B). (c) Using Proposition 6.1.1(d), we have P(A B) = IAEBBI = IAuBI-IAnBI = IAUB _IAnBI EB lSI lSI lSI lSI = P(A U B) -P(A n B) = (P(A) + P(B) -P(A n B)) -P(A n B) = P(A) + P(B) -2P(A n B). (d) [BB] From (c), P(A EB B) P(A) + P(B) -2P(A n B) = P(A) + P(B) -2(P(A) + P(B) -P(A U B)) = 2P(A U B) -P(A) -P(B). (e) [BB] P(A n (B U e)) = P((A n B) U (A n e)) = P(A n B) + P(A n e) -P(A n B n e). (t) Using (e), P(A n (B U e)) = (P(A) + P(B) -P(A U B)) + (P(A) + p(e) -P(A U e)) -P(A n B n e) = 2P(A) + P(B) + p(e) -P(A U B) -P(A U e) -P(A n B n e). (g) P(A U (B n e) P((A U B) n (A U e)) = P(A U B) + P(A U e) -P(A U B U e)....
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