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ics141-lecture03-Logic

# ics141-lecture03-Logic - 3-1ICS 141 Discrete Mathematics...

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Unformatted text preview: 3-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga3-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 3Chapter 1. The Foundations1.2 Propositional Equivalences1.3 Predicates and Quantifiers3-3ICS 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. Baek3-4ICS 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 p↔q 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.3-5ICS 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: Equivalences3-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiEquivalence Laws - ExamplesIdentity: p∧T ⇔p p∨F ⇔pDomination: p∨T ⇔T p∧F ⇔FIdempotent: p∨p ⇔p p∧p ⇔pDouble negation: ¬¬p ⇔pCommutative: p∨q ⇔q∨p p∧q ⇔q∧pAssociative: (p∨q)∨r⇔p∨(q∨r)(p∧q)∧r⇔p∧(q∧r)Topic #1.1 – Propositional Logic: Equivalences3-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMore Equivalence LawsDistributive: p∨(q∧r) ⇔(p∨q)∧(p∨r)p∧(q∨r) ⇔(p∧q)∨(p∧r)De Morgan’s:¬(p∧q) ⇔¬p ∨¬q¬(p∨q) ⇔¬p ∧¬qAbsorptionp∨(p∧q) ⇔pp∧(p∨q) ⇔pTrivial tautology/contradiction:p∨¬p⇔Tp∧¬p⇔FTopic #1.1 – Propositional Logic: EquivalencesSee Table 6, 7, and 8 of Section 1.23-8ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiDefining Operators via EquivalencesUsing equivalences, we can defineoperators in terms of other operators.Exclusive or: p⊕q⇔(p∨q)∧¬(p∧q)p⊕q⇔(p∧¬q)∨(q∧¬p)Implies: p→q ⇔¬p ∨qBiconditional: p↔q ⇔(p→q)∧(q→p)p↔q ⇔¬(p⊕q)3-9ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiAn Example ProblemCheck using a symbolic derivation whether (p ∧¬q) →(p⊕r)⇔¬p ∨q∨¬r....
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ics141-lecture03-Logic - 3-1ICS 141 Discrete Mathematics...

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