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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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 Spring '11
 Shirley
 Logic, Rational number

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