ics141-lecture04-Logic

ics141-lecture04-Logic - University of Hawaii ICS141:...

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4-1 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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4-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 4 Chapter 1. The Foundations 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference
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4-3 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 material was done by Prof. Baek
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4-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii The Universal Quantifier 2200 2200 x P ( x ) is true if P ( x ) is true for every x in D, false if P ( x ) is false for at least one x in D For every real number x , x 2 ≥ 0 : true For every real number x , x 2 –1 > 0 : false A counterexample to the statement 2200 x P ( x ) : a value x in D that makes P ( x ) false If all the elements in the domain can be listed as x 1 , x 2 ,…, x n then, 2200 x P ( x ) is the same as the conjunction P ( x 1 ) P ( x 2 ) ··· P ( x n ) Topic #3 – Predicate Logic
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4-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii The Existential Quantifier 5 5 x P ( x ) is true if P ( x ) is true for at least one x in D, false if P ( x ) is false for every x in D If all the elements in the domain can be listed as x 1 , x 2 ,…, x n then, 5 x P ( x ) is the same as the disjunction P ( x 1 ) P ( x 2 ) ··· P ( x n ) Topic #3 – Predicate Logic
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4-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Quantifiers with Restricted Domains Sometimes the universe of discourse is restricted within the quantification, e.g. , 2200 x> 0 P ( x ) is shorthand for “For all x that are greater than zero, P ( x ).” = 2200 x ( x> 0 P ( x )) 5 x> 0 P ( x ) is shorthand for “There is an x greater than zero such that P ( x ).” = 5 x ( x> 0 P ( x )) Topic #3 – Predicate Logic
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4-7 University of Hawaii Quantifier Equivalence Laws Definitions of quantifiers: If u.d.=a,b,c,… 2200 x P ( x ) P (a) P (b) P (c) 5 x P ( x ) P (a) P (b) P (c) From those, we can prove the laws: 2200 x P ( x ) ¬ ( 5 x ¬ P ( x )) 5 x P ( x )
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ics141-lecture04-Logic - University of Hawaii ICS141:...

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