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Set 3.1 Questions - 1 06 Chapter 3 The Logic of Quantified...

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Exercise Set 3.1 * 1. A menagerie consists of seven brown dogs, two black dogs, six gray cats, ten black cats, five blue birds, six yellow birds, and one black bird. Determine which of the following state- ments are true and which are false. a. There is an animal in the menagerie that is red. b. Every animal in the menagerie is a bird or a mammal. c. Every animal in the menagerie is brown or gray or black. d. There is an animal in the menagerie that is neither a cat nor a dog. e. No animal in the menagerie is blue. f. There are in the menagerie a dog, a cat, and a bird that all have the same color. 2. Indicate which of the following statements are true and which are false. Justify your answers as best as you can. 3. Let P(x) be the predicate "x > l/x." a. Write P(2), P(~), P(-I), P(-~), and P(-8), and indicate which of these statements are true and which are false. b. Find the truth set of P (x) if the domain of x is R, the set of all real numbers. c. If the domain is the set R+ of all positive real numbers, what is the truth set of P(x)? 4. Let Q(n) be the predicate "n 2 ::: 3D." a. Write Q(2), Q(-2), Q(7), and Q(-7), and indicate which of these statements are true and which are false. b. Find the truth set of Q(n) if the domain of n is Z, the set of all integers. c. If the domain is the set Z+ of all positive integers, what is the truth set of Q(n)? 5. Let Q(x, y) be the predicate "If x < y then x 2 < y2" with domain for both x and y being the set R of real numbers. Q(x, y) is false. Q(x, y) is true. 6. Let R(m, n) be the predicate "If m is a factor of n 2 then m is a factor of n," with domain for both m and n being the set Z of integers. a. Explain why R(m, n) is false if m = 25 and n = 10. b. Give values different from those in part (a) for which R(m, n) is false. c. Explain why R(m, n) is true if m = 5 and n = 10. d. Give values different from those in part (c) for which R(m, n) is true.
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