Chapter2 - Chapter 2 The Logic of Quantified Statements...

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Chapter 2: The Logic of Quantified Statements January 14, 2008
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Outline 1 2.1 Introduction to Predicates and Quantified Statements I 2 2.2 Introduction to Predicates and Quantified Statements II 3 2.3 Statements Containing Multiple Quantifiers
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Consider the following argument All students in Electrical Engineering have to take MTH314. Tarik is a student in Electrical Engineering. Tarik has to take MTH314. This seems to be a valid argument from the point of view of logic. However, we cannot show that using the methods of Chapter 1, since the first statement cannot be translated using only the symbols , , , , or .
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Basic Facts and Notation For Sets A set is some collection of objects, which we call elements of the set. For example, A = { blue , red , yellow } and B = { 0 , 2 , 5 , 13 , - 4 } are examples of sets. To write the fact that an object x is an element of a set A , we write x A while x A will denote the fact that x is not an element of A .
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We will use the following notation for familiar sets of numbers: 1 R for the set of all real numbers 2 Z for the set of all integers 3 Q for the set of all rational numbers (quotients of integers) By putting + or - in the superscript, we indicate the set of positive or negative numbers; e.g R + will stand for all positive real numbers, etc. Instead of writing Z + , the set of all positive integers is often called the set of natural numbers and the notation for it is N .
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Predicates We have seen in Chapter 1, that a sentence x 2 + x > 2 is not a statement since its truth (or falsity) depends on what value we assign to the variable x . Similarly, the sentence She is a student at Ryerson. is not a statement either, since we have to know who “she” refers to in order to determine the truth value. Such sentences are called predicates . They are not statements unless we interpret the variables in them as particular elements of some prescribed set. After that, they become true or false statements.
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Definition A predicate is a sentence that contains a finite number of variables and becomes a statement once these variables are assigned some specific values. The domain of a predicate variable is the set of all values that can be substituted for the variable. For example, for the predicate P ( x ) x 2 + x > 2 the domain can be any set of numbers ( R , Z , . . . ) in which its operations (+, · ,. . . ) make sense. A predicate could involve any number of variables; e.g. Q ( x , y ) is the sentence x is divisible by y 3
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Example Consider the predicate P ( x ) x 2 + x > 2 with the domain R . For what x R is this predicate true? For x = 2, P ( 2 ) : 2 2 + 2 > 2 , true For x = - 1 3 , P ( - 1 3 ) : ( - 1 3 ) 2 + ( - 1 3 ) > 1 , false It can be shown, using some basic facts about quadratic equations that P ( x ) is true provided x < - 2 or x > 1
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Definition If P ( x ) is a predicate whose truth domain is D , the truth set of P ( x ) is the set of all elements of D which make P ( x ) true when they are substituted for x . We write the truth set for P ( x ) as { x D | P ( x ) } which we read as “the set of all x in D such that P ( x ) is true”.
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Example Consider the predicate Q ( n )
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