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Unformatted text preview: CPSC 121 Lecture 9 January 23, 2009 Menu January 23, 2009 Topics: Predicates and Quantified Statements (cont’d) (Supplementary) Examples Reading: Today: Epp 2.1 Next week: Epp 2.3, 2.2, 2.4 Reminders: Assignment 1 due Friday, January 30, 17:00 Inclass Quiz 1 Wednesday, February 4 Midterm exam Tuesday, February 24 (evening) READ the WebCT Vista course announcements board Predicates and Quantified Statements (cont’d) Recall, from last lecture, the definition and meaning of the term predicate... Recap: Definition of Predicate Let P ( x ) be a statement involving a variable x . Let D be the set of values that x can take on Definition: P ( x ) is a predicate with respect to D if for each x ∈ D , P ( x ) is a proposition D is called the domain of discourse for x Remember: For P ( x ) to be a predicate, it must be a proposition for every x ∈ D . Now, let’s consider ways to combine predicates. Combining Predicates As with propositions, we use logical connectives to combine predicates to make new predicates Let P ( x ) : x > 3 and Q ( x ) : x < 10 be predicates. Then, we can define a new predicate, R ( x ) , as: R ( x ) ≡ P ( x ) ∧ Q ( x ) NOTE: We do need to specify the domain of discourse, D . Here, we might say that D = R , the set of real numbers There is another important way to create propositions from predicates called quantification Universal Quantification Consider the mathematical statement: ( x + 1)( x 1) = x 2 1 Question: What words might we add to make this statement a true proposition? Answer: We might write, in English, “For all real numbers x , ( x + 1)( x 1) = x 2 1 ” In logic we write, ∀ x ∈ R , ( x + 1)( x 1) = x 2 1 The symbol, ∀ , is called the universal quantifier Universal Quantification (cont’d) Definition: Let P ( x ) be a predicate with respect to a domain of discourse,...
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 Spring '08
 BELLEVILLE
 Logic, Semantics, Proposition, Universal quantification, ∀x, universally quantified statement

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