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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 2.1 Propositional Functions and Quantified Statements In Chapter 1, we defined a statement, or a proposition, as a declarative sentence which has a definite truth value. Most sentences which we encounter in mathematics are not statements, however, because they involve one or more variables, with a truth value which depends upon the value or values of the variables. In this chapter, we consider such sentences, which are called propositional functions . Example. Identify the variables in each of the following sentences and determine values of the variable for which the sentence is true and values for which it is false. (a) 2 x + 1 = 6 Solution. This sentence contains the variable x . Assuming x represents a real number, then the sentence is true when x = 5 2 and is false when x is any other real number. When x = 5 2 , the sentence becomes 2( 5 2 ) + 1 = 6 , which is a true statement. For another real number, say x = 4, the sentence becomes 2( 4) + 1 = 6 , which is a false statement. (b) 0 x < y 2 Solution. This contains two variables, x and y . Assuming both x and y represent real numbers, when, for instance, x = 4 and y = 3, the sentence becomes 4 < ( 3) 2 , which is a true statement. When, for instance, x = 15 and y = 3, the sentence becomes 15 < 3 2 , which is a false statement. (c) The function f is continuous on the interval [ 1 , 1]. Solution. This sentence contains the variable f , which represents a function. When, for instance, f is the function defined by f ( x ) = x 2 for all real numbers x , then the sentence becomes The function defined by f ( x ) = x 2 is continuous on the interval [ 1 , 1]. This is a true statement. When, for instance, f is the function defined for all real numbers x by f ( x ) = 1 if x < 1 if x then the sentence is a false statement. (d) The sets A and B have no elements in common. Solution. This sentence contains the variables A and B , where A and B represent sets. When A = { 1 , 2 , 3 } and B = { 5 , 6 , 7 , 8 } , then the sentence is a true statement: The sets A = { 1 , 2 , 3 } and B = { 5 , 6 , 7 , 8 } have no elements in common. When A = { 1 , 2 , 3 , 4 } and B = { 2 , 4 , 6 , 8 } , then the sentence is a false statement. 31 32 Chapter 2 Predicate Logic The sentences in the above example are examples of propositional functions, defined as follows. Definition. A propositional function , or predicate , is a declarative sentence that contains one or more variables which becomes a statement when specific values are substituted for the variables. The set of values which may be substituted for a variable is called its domain ....
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 Spring '08
 Solheid
 Math

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