08a-Predicates-Quantifiers-II

08a-Predicates-Quantifiers-II - PIntroduction Quantifiers...

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Predicates and Quantifiers II Discrete Mathematics
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Discrete Mathematics – Predicates and Quantifiers II 8-2 Previous Lecture Predicates Assigning values, universe, truth values Quantifiers Representing statements using quantifiers
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Discrete Mathematics – Predicates and Quantifiers II 8-3 How to Write Formulas Correctly: Syntax Already saw: Predicates, Quantifiers Often predicates have more than one variable. In this case we need more than one quantifier. P(x,y) = ``car x has colour y’’ x y P(x,y) ``every car is painted all colours x y P(x,y) ``there is a car that is painted some colour x y P(x,y) ``every car is painted some colour x y P(x,y) ``there is a car that is painted all colours
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Discrete Mathematics – Predicates and Quantifiers II 8-4 How to Write Formulas Correctly: Syntax Quantifiers can be used together with logic connectives ``Every car is either red or blue’’ P(x) - ``car x is red’’ Q(x) - ``car x is blue’’ x (P(x) Q(x)) ``Everyone who knows a current password can logon onto the network’’ P(x) - ``x knows a current password’’ Q(x) - ``x can logon onto the network’’ x (P(x) Q(x))
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Discrete Mathematics – Predicates and Quantifiers II 8-5 How to Write Formulas Correctly: Syntax (cntd) Logic connectives can be put between quantified statements ``Every car is blue, or there is a red car’’ P(x) - ``car x is blue’’ Q(x) - ``car x is red’’ ( x P(x)) ( x Q(x)) ``For every number there is a smaller one, or there is the least number’’ We use predicate x y ( x y (y x)) ( x y (x y))
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Discrete Mathematics – Predicates and Quantifiers II 8-6 Examples of Formulas: Rules and Laws Predicates and quantifiers are (implicitly) present in rules and laws ``Everyone having income more that $20000 must file a tax report’’ P(x) - ``x has income more than $20000’’ Q(x) - ``x must file a tax report’’ x (P(x) Q(x))
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Discrete Mathematics – Predicates and Quantifiers II 8-7 Examples of Formulas: Theorems and Math Statements Every theorem involves predicates and quantifiers ``For every statement there is an equivalent CNF’’ C(x) - ``x is a CNF’’ x y (C(y) (x y)) ``A parallelogram is a rectangle if all its angles are equal’’ R(x) - ``parallelogram x is a rectangle’’ A(x) - ``all angles of x are equal’’ x (A(x) R(x))
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Discrete Mathematics – Predicates and Quantifiers II 8-8 Semantics: Meaning of the Formulas.
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This note was uploaded on 12/12/2010 for the course MACM 201 taught by Professor Marnimishna during the Fall '09 term at Simon Fraser.

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08a-Predicates-Quantifiers-II - PIntroduction Quantifiers...

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