Discrete Mathematics with Graph Theory (3rd Edition) 198

Discrete Mathematics with Graph Theory (3rd Edition) 198 -...

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196 Solutions to Exercises 20. [BB] We use induction on n, the strong form. For n = 1, the result is clear. If n = 2, P(AI U A2) = P(AI) + P(A2) - P(AI n A2) ~ P(AI) + P(A2)' Now assume n > 2 and the result holds for all integers k with 1 ~ k < n. We have P(AI U A2 U . .. U An) = P((AI U A2 U· U An-I) U An) ~ P(AI U A2 U . .. U An-I) + P(An) ~ (P(AI) + P(A2) + . .. + P(An- I) + P(An) By the Principle of Mathematical Induction, the result follows. using the k = 2 case using the k = n - 1 case. 21. We use induction on n, the strong form. For n = 1, the result is clear. If n = 2, P(AI n A2) = P(AI) + P(A2) - P(AI U A2) ~ P(AI) + P(A2) - 1. Now assume n > 2 and the result holds for all integers k with 1 ~ k < n. We have P(AI n A2 n . .. nAn) = P((AI n A2 n· n An-I} nAn) ~ P(AI n A2 n . .. nAn-I) + P(An) - 1 ~ (P(AI) + P(A2) + . .. + P(An- I) - (n - 2) + P(An) - 1 = P(AI) + P(A2) + . .. + P(An) - (n - 1).
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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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