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Unformatted text preview: Chapter 3: The Logic of Quantified Statements Section 3.1 6. a . When m = 25 and n = 10 , the statement m is a factor of n 2 is true because n 2 = 100 and 100 = 4 25 . But the statement m is a factor of n is false because 10 is not a product of 25 times any integer. Thus the hypothesis is true and the conclusion is false, so the statement as a whole is false. b . R ( m,n ) is also false when m = 8 and n = 4 because 8 is a factor of 4 2 = 16, but 8 is not a factor of 4. c. When m = 5 and n = 10, both statements m is a factor of n 2 and m is a factor of n are true because n = 10 = 5 2 = m 2 and n 2 = 100 = 5 20 = m 20. Thus both the hypothesis and conclusion of R ( m,n ) are true, and so the statement as a whole is true. d . Here are examples of two kinds of correct answers: (1) Let m = 2 and n = 6. Then both statements m is a factor of n 2 and m is a factor of n are true because n = 6 = 2 3 = m 3 and n 2 = 36 = 2 18 = m 18. Thus both the hypothesis and conclusion of R ( m,n ) are true, and so the statement as a whole is true. (2) Let m = 6 and n = 2. Then both statements m is a factor of n 2 and m is a factor of n are false because n = 2 6 = 6 k , for any integer k , and n 2 = 4 6 = 6 j , for any integer j. Thus both the hypothesis and conclusion of R ( m,n ) are false, and so the statement as a whole is true. 12. Counterexample : Let x = 1 and y = 1, and note that x + y = 1 + 1 = 2 whereas x + y = 1 + 1 = 1 + 1 = 2 , and 2 6 = 2 . (This is one counterexample among many. Any real numbers x and y with xy 6 = 0 will produce a counterexample.) 15. a. Some acceptable answers: All rectangles are quadrilaterals. If a figure is a rectangle then that figure is a quadrilateral. Every rectangle is a quadrilateral. All figures that are rectangles are quadrilaterals. Any figure that is a rectangle is a quadrilateral. b . Some acceptable answers: There is a set with sixteen subsets. Some set has sixteen subsets. Some sets have sixteen subsets. There is at least one set that has sixteen subsets. 18. c . s , if C ( s ) then E ( s ) . d . x such that C ( s ) M ( s ) . 21. b . The base angles of T are equal, for any isosceles triangle T . d . f is not differentiable, for some continuous function f . 24. b . a question x such that x is easy. x such that x is a question and x is easy. 27. c. This statement translates as There is a square that is above d . This is false because the only objects above d are a (a triangle) and b (a circle)....
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This note was uploaded on 05/10/2011 for the course MATH 325K taught by Professor Shirley during the Spring '11 term at University of Texas at Austin.
- Spring '11