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02 Predicate Logic

02 Predicate Logic - ENGG1007 Foundations of Computer...

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1 ENGG1007 Foundations of Computer Science Predicate Logic Predicate Logic Prof. Francis Chin, Dr SM Yiu (chapters 1.3, 1.4)
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2 ENGG1007 FCS Subjects and Predicates Subjects and Predicates Do you still remember the definition of propositions? Is “x > 3” a proposition? No, unless the value of x is fixed. Let P(x) denote the statement “x > 3” where P(x) is called a propositional function . P(x) has a truth value once the value of x is fixed. E.g. P(5) is true; P(1) is false. P actually refers to the property “is greater than 3” and is called the predicate . Note that x is a variable (the subject). A propositional function can have more than one variable (multi- value predicates). E.g. Let Q(x, y) denote “x + y > 10”. Then, Q(4, 5) is false; Q(5, 6) is true.
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3 ENGG1007 FCS Universe of Discourse Universe of Discourse The collection of values that a variable x may take is called the universe of discourse . Example : “x is rich”, x can refer to people in HK, the world, movies stars, IT people, … For the statement “x is prime ”, the universe of discourse of x is the set of all positive integers.
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4 ENGG1007 FCS Quantification (Quantifiers) Quantification (Quantifiers) Two ways to convert a propositional function, say P(x), into a proposition. Assign value(s) to variable(s). Quantify variable(s): P(x) is true for all possible values of x (Universal Quantification) These exists at least one value of x such that P(x) is true (Existence Quantification) Example: Let P(x) denote the statement “x has 10 girl friends”. P(x) has no truth value and is not a proposition. P(SM Yiu) is a proposition . 2200 x P(x) is a proposition. “for all possible values of x (universe of discourse), P(x) is true” e.g. 2200 x P(x)” is false where x is any lecturer in our dept.
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5 ENGG1007 FCS Example: Let P(x) denote the statement “x is married”.
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