ics141-lecture05-Proofs

ics141-lecture05-Proofs - 5-1ICS 141: Discrete Mathematics...

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Unformatted text preview: 5-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga5-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 5Chapter 1. The Foundations1.5 Rules of Inference1.6 Introduction to Proofs5-3ICS 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 made by Prof. Baek5-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiValid ArgumentsA form of logical argument is validif whenever every premise is true, the conclusion is also true. A form of argument that is not valid is called a fallacy.5-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiInference Rules: General FormAn Inference Ruleis A pattern establishing that if we know that a set of premisestatements of certain forms are all true, then we can validly deduce that a certain related conclusionstatement is true. premise 1premise 2 ···.∴conclusion “∴” means “therefore”5-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiInference Rules & ImplicationsEach valid logical inference rule corresponds to an implication that is a tautology.premise 1 premise 2 Inference rule···.∴conclusionCorresponding tautology: ((premise 1) ∧(premise 2) ∧…) →conclusion5-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiModus Ponensp Rule of modus ponensp→q(a.k.a. law of detachment)∴q(p ∧(p→q)) →q is a tautology“the mode of affirming”pqp →qp ∧(p→q)(p ∧(p→q)) →qTTTTTTFFFTFTTFTFFTFTNotice that the first row is the only one where premises are all true 5-8ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiModus Ponens: Examplep→q : “If it snows today then we will go skiing”p .: “It is snowing today”∴q : “We will go skiing”p→q : “If nis divisible by 3 then n2is divisible by 3”p .: “nis divisible by 3”∴q : “n2is divisible by 3”IfThenAssumed TRUEis TRUEIfThenAssumed TRUEis TRUE5-9ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiModus Tollens¬q Rule of modus tollens p→q∴¬p(¬q ∧(p→q)) →¬p is a tautologyExamplep→q : “If this jewel is really a diamond then it will scratch glass”¬q .: “The jewel doesn’t scratch glass”∴¬p : “The jewel is not a diamond”“the mode of denying”IfAssumed TRUEThenis TRUE5-10ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMore Inference RulespRule of Addition∴p∨q p →(p∨q)p∧qRule of Simplification∴p p∧q →ppqRule of Conjunction∴p∧q (p)∧(q) →p∧q 5-11ICS 141: Discrete Mathematics I (Spr 2008)University of Hawaii...
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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.

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ics141-lecture05-Proofs - 5-1ICS 141: Discrete Mathematics...

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