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08 - Predicates and Quantifiers II Introduction Discrete...

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Introduction Predicates and Quantifiers II Discrete Mathematics Andrei Bulatov
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Discrete Mathematics – Predicates and Quantifiers II 8-2 Previous Lecture Predicates Assigning values, universe, truth values
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Discrete Mathematics – Predicates and Quantifiers II 8-3 Relational Databases A relational database is a collection of tables like No. Name Student ID Supervisor Thesis title 1. Bradley Coleman 30101234 Petra Berenbrink Algebraic graph theory A table consists of a schema and an instance. A schema is a collection of attributes, where each attribute has an associated universe of possible values. An instance is a collection of rows, where each row is a mapping that associates with each attribute of the schema a value in its universe. Every table is a predicate that is true on the rows of the instance and false otherwise.
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Discrete Mathematics – Predicates and Quantifiers II 8-4 Quantifiers One way to obtain a statement from a predicate is to assign all its variables some values Another way to do that is to use expressions like `For every …’ `There is … such that …’ `A … can be found …’ `Any … is …’ `Every man is mortal’ `There is x such that x is greater than 3’ `There is a person who is my father’ `For any x , ' 0 2 x quantification
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Discrete Mathematics – Predicates and Quantifiers II 8-5 Universal Quantifiers Abbreviates constructions like For all … For any … Every … Each … Asserts that a predicate is true for all values from the universe `Every man is mortal’ `All lions are fierce’ Notation: 2200 `For any x , ' 0 2 x 2200 x P(x) means that for every value a from the universe P(a) is true
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Discrete Mathematics – Predicates and Quantifiers II 8-6 Universal Quantifiers (cntd) 2200 x P(x) is false if and only if there is at least one value a from the universe such that P(a) is false Such a value a is called a counterexample Thus to disprove that `Every man is mortal’ it suffices to recall the movie `Highlander’ `For any x , ' 0 2 x true!
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