Discrete Structure lecture_7

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 # 7SET IDENTITIESKHAWAJA TARIQ MAHMOODLecture # 7SET IDENTITIESSET IDENTITIES:Let A, B, C be subsets of a universal set U.1.Idempotent Lawsa.2.4.5.6.7.b.AA=Ab.AB=BACommutative Lawsa.3.AA=AAB=BAAssociative Lawsa.A (B C) = (A B) Cb.A (B C) = (A B) CDistributive Lawsa.A (B C) = (A B) (A B)b.A (B C) = (A B) (A C)Identity Lawsa.A=Ac.AU=Ub. A = d. A U = AComplement Lawsa.A Ac = Uc.Uc = b. A Ac = d. c = UDouble Complement Law(Ac) c = A8.De-Morgans Lawsa. (A B)c = Ac Bc9.b. (A B)c = Ac BcAlternative Representation for Set DifferenceKhawaja Tariq Mahmood.Page 2Lecture # 7SET IDENTITIESA B = A Bc10.Subset Laws11.a.A B C iff A C and B Cb.C A B iff C A and C BAbsorption Lawsa. A (A B) = Ab. A (A B) = AEXAMPLE:Prove the De-Morgans Law: A B Ac B ccPROOF:Letx A Bx A Bx Aandcx Aandcx A Bc A BccxBx Bc Ac B c i ConverselyLety Ac B cy Ac andy A andy A By A By BcyBcAc B c A B

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