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ics141-lecture05-Proofs

ics141-lecture05-Proofs - University of Hawaii ICS141...

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5-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga
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5-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 5 Chapter 1. The Foundations 1.5 Rules of Inference 1.6 Introduction to Proofs
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5-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some 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. Baek
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5-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Valid Arguments A form of logical argument is valid if whenever every premise is true, the conclusion is also true. A form of argument that is not valid is called a fallacy .
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5-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Inference Rules: General Form An Inference Rule is A pattern establishing that if we know that a set of premise statements of certain forms are all true, then we can validly deduce that a certain related conclusion statement is true. premise 1 premise 2 ··· . conclusion ” means “therefore”
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5-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Inference Rules & Implications Each valid logical inference rule corresponds to an implication that is a tautology. premise 1 premise 2 Inference rule ··· . conclusion Corresponding tautology: (( premise 1 ) ( premise 2 ) …) conclusion
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5-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Modus Ponens p Rule of modus ponens p q (a.k.a. law of detachment ) q ( p ( p q )) q is a tautology “the mode of affirming” p q p q p ( p q ) ( p ( p q )) q T T T T T T F F F T F T T F T F F T F T Notice that the first row is the only one where premises are all true
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5-8 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Modus Ponens: Example p q : “If it snows today then we will go skiing” p . : “It is snowing today” q : “We will go skiing” p q : “If n is divisible by 3 then n 2 is divisible by 3” p . : “ n is divisible by 3” q : “ n 2 is divisible by 3” If Then Assumed TRUE is TRUE If Then Assumed TRUE is TRUE
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5-9 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Modus Tollens ¬ q Rule of modus tollens p q ∴¬ p ( ¬ q ( p q )) ¬ p is a tautology Example p 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” If Assumed TRUE Then is TRUE
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5-10 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii More Inference Rules p Rule of Addition p q p ( p q ) p q Rule of Simplification p p q p p q Rule of Conjunction p q ( p ) ( q ) p q
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