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# 08h - Discrete Mathematics Predicates and Quantifiers II...

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Introduction Predicates and Quantifiers II Discrete Mathematics Andrei Bulatov Discrete Mathematics – Predicates and Quantifiers II 8-2 Previous Lecture Predicates Assigning values, universe, truth values Quantifiers Discrete Mathematics – Predicates and Quantifiers II 8-3 Quantifiers and Negations Summarizing true false 2200 x P(x) 5 x P(x) Observe that 2200 x P(x) is false if and only if 5 x ¬ P(x) is true 5 x P(x) is false if and only if 2200 x ¬ P(x) is true For every value a from the universe P(a) is true There is a counterexample – a value a from the universe such that P(a) is false There is a value a from the universe such that P(a) is true For all values a from the universe P(a) is false Discrete Mathematics – Predicates and Quantifiers II 8-4 Example What is the negation of each of the following statements? Statement Negation All lions are fierce 2200 x P(x) There is a peaceful lion Everyone has two legs 2200 x P(x) There is a person having more than two legs, one leg, or no legs at all Some people like coffee 5 x P(x) All people hate coffee There is a lady in one of these rooms 5 x P(x) There is a tiger in every room (Some rooms contains a lady) Discrete Mathematics – Predicates and Quantifiers II 8-5 Multiple 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’’ 2200 x 2200 y P(x,y) ``every car is painted all colours 5 x 5 y P(x,y) ``there is a car that is painted some colour 2200 x 5 y P(x,y) ``every car is painted some colour 5 x 2200 y P(x,y) ``there is a car that is painted all colours Discrete Mathematics – Predicates and Quantifiers II 8-6 Open and Bound Variables In the statement ``car x has some colour’’ 5 y P(x,y) variables x and y play completely different roles. Variable y is bound by the existential quantifier. Effectively it disappeared from the statement. Variable x is not bound, it is free . Another example: ``x is the least number’’ Q(x,y) = ``x is less than y’’ 2200 y Q(x,y) or 2200 y (x y) ``x is the greatest number’’ 2200 y Q(y,x) or 2200 y (y x)

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Discrete Mathematics – Predicates and Quantifiers II 8-7 Defining New Predicates Quantifying some variables helps creating new predicates ``car x has some colour’’ Q(x) = 5 y P(x,y) Let P(x,y) mean ``lion x likes y’’ Q(y) = 2200 x P(x,y) ``every lion likes y’’ Q(meat) is true Q(apples) is false R(x) = 2200 y P(x,y) ``lion x likes everything’’ R(x) is always false (say it using quantifiers: 2200 x ¬ R(x) or 2200 x ¬ ( 2200 y P(x,y)) ) S(x) = 5 y P(x,y) ``lion x has favorite food’’ Discrete Mathematics – Predicates and Quantifiers II 8-8 Quantifiers and Compound Statements Quantifiers can be used together with logic connectives
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08h - Discrete Mathematics Predicates and Quantifiers II...

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