Discrete Structure lecture_28

Apr 28th, 2015
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It includes all the details about the Discrete structure with examples.Step by step solution.

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DISCREAT STRUCTURELECTURE # 28THE INCLUSIONEXCLUSIONPRINCIPLEKHAWAJA TARIQ MAHMOODLecture # 28THE INCLUSION-EXCLUSION PRINCIPLETHE INCLUSION-EXCLUSION PRINCIPLE:1 .If A and B are disjoint finite sets, thenn(AB) = n(A) + n(B)2.If A and B are finite sets, thenn(AB) = n(A) + n(B) - n(AB)EXAMPLE:There are 15 girls students and 25 boys students in a class. How many students arethere in total?SOLUTION:Let G be the set of girl students and B be the set of boy students.Thenn(G) = 15;andn(G B) = ?n(B) = 25Since, the sets of boy and girl students are disjoint; here total number of students aren(G B)= n(G) + n(B)= 15 + 25= 40EXAMPLE:Among 200 people, 150 either swim or jog or both. If 85 swim and 60 swim and jog,how many jog?SOLUTION:Let U be the set of people considered. Suppose S be the set of people who swim andJ be the set of people who jog. Then given n(U) = 200;n(S J) = 150Khawaja Tariq Mahmood.Page 2Lecture # 28THE INCLUSION-EXCLUSION PRINCIPLEn(S J) = 60n(S) = 85;and n(J) = ?By inclusion - exclusion principle,n(S J) = n(S) + n(J) - n(S J)150 = 85 + n(J) - 60n(J) = 150 - 85 + 60= 125Hence 125 people jog.EXAMPLE:Let A and B are subset of U with n(U) = 100, n(A) = 50, n(B)= 60, and n((AB))= 20. Find n(AB)SOLUTION:Since (A B) = U \ (A B)n ((A B)) = n (U) - n (A B)20n (A B)= 100 - n (A B)= 100 - 20 = 80Now, by

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