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Unformatted text preview: 31ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga32ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 3Chapter 1. The Foundations1.2 Propositional Equivalences1.3 Predicates and Quantifiers33ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of Discrete Mathematics and Its Applications written by Kenneth H. Rosen.Some slides were done by Prof. Baek34ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLogical EquivalenceCompound proposition pis logically equivalentto compound proposition q, written p q or p q, iffthe compound proposition pq is a tautology.Compound propositions pand q are logically equivalent to each other iffpand q contain the same truth values as each other in allrows of their truth tables.35ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiEquivalence LawsThese are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead.They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it.Topic #1.1 Propositional Logic: Equivalences36ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiEquivalence Laws  ExamplesIdentity: pT p pF pDomination: pT T pF FIdempotent: pp p pp pDouble negation: p pCommutative: pq qp pq qpAssociative: (pq)rp(qr)(pq)rp(qr)Topic #1.1 Propositional Logic: Equivalences37ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMore Equivalence LawsDistributive: p(qr) (pq)(pr)p(qr) (pq)(pr)De Morgans:(pq) p q(pq) p qAbsorptionp(pq) pp(pq) pTrivial tautology/contradiction:ppTppFTopic #1.1 Propositional Logic: EquivalencesSee Table 6, 7, and 8 of Section 1.238ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiDefining Operators via EquivalencesUsing equivalences, we can defineoperators in terms of other operators.Exclusive or: pq(pq)(pq)pq(pq)(qp)Implies: pq p qBiconditional: pq (pq)(qp)pq (pq)39ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiAn Example ProblemCheck using a symbolic derivation whether (p q) (pr)p qr....
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.
 Fall '08
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