At least one of us is a knave. What are A and B?

Transcription

1 1. This is a puzzle about an island in which everyone is either a knight or a knave. Knights always tell the truth and knaves always lie. This problem is about two people A and B, each of whom is either a knight or a knave. A made the following statement: At least one of us is a knave. What are A and B? (a) A and B are both knights. (b) A is a knight and B is a knave. (c) A is a knave and B is a knight. (d) A and B are both knaves. (e) None of the above. 1

2 2. Let p and q be the propositions: p: It is below freezing. q: It is snowing. Which of the following expresses the statement: It is either snowing or below freezing. (a) p q (b) p q (c) p q (d) None of the above 2

3 3. Let p and q be the propositions: p: It is below freezing. q: It is snowing. Which of the following expresses the statement: It is snowing but not below freezing. (a) p q (b) q p. (c) p q. (d) p q (e) None of the above (it is impossible to be above freezing and snowing). 3

4 4. On the island of knights and knaves, suppose A says, Either I am a knave or B is a knight. What are A and B? (a) A and B are both knights. (b) A is a knight and B is a knave. (c) A is a knave and B is a knight. (d) A and B are both knaves. (e) None of the above. 4

5 5. Determine whether the following statement is true or false: If = 3 then = 4. (a) True (b) False (c) Part of it is true and part of it is false. (d) Can t decide, since the statement is nonsense. 5

6 6. Choose the statement that describes you best. (a) I looked at the course website and the textbook. (b) I looked at the course website but not the textbook. (c) I looked at the textbook but not the course website. (d) I haven t looked at either, but I will before next class. 6

7 7. What is my name? (a) Dr. Ford (b) Dr. Doedel (c) Dr. Narayanan (d) Don t know 7

8 8. p q (p q) ( p q) (a) is a true statement. (b) is a false statement. (c) is sometimes true, sometimes false. (d) don t know. 8

9 9. Let the universe of discourse be all integers. Then the truth value of x x 3 > x is: (a) False (b) True (c) Sometimes true, sometimes false (d) Don t know 9

10 10. Which of the two parts in Q 3 of worksheet 2 are logically equivalent? (a) Parts (c) and (d) are equivalent. (b) Parts (b) and (c) are equivalent. (c) Parts (e) and (f) are equivalent. (d) Parts (c) and (f) are equivalent. (e) Parts (a) and (e) are equivalent. 10

11 11. The negation of the statement x P (x) Q(x) is (a) x (P (x) Q(x)) (b) x P (x) Q(x) (c) x Q(x) P (x) (d) x P (x) Q(x) (e) Don t know. 11

12 12. Let P (x) denote x drinks tea and Q(x) denote x does not drink coffee. Let the universe of discourse be all Concordia students. Then the statement x P (x) Q(x) means (a) All Concordia students drink either tea or coffee. (b) There is a Concordia student who drinks tea but not coffee. (c) All Concordia students who drink coffee also drink tea. (d) All Concordia students drink both coffee and tea. (e) All Concordia students who drink tea also drink coffee. 12

13 13. How am I doing in terms of pace? (a) Too fast (b) Too slow (c) About right 13

14 14. Let the universe of discourse be integers and let P (x) denote the predicate x is the square root of 2. Then the statement x P (x) means (a) There is no integer which is the square root of 2. (b) There is an integer which is not the square root of 2. (c) Every integer is not the square root of 2. (d) There is an integer which is the square root of 2. (e) Don t know. 14

15 15. Let the universe of discourse be integers and let P (x) denote the predicate x is the square root of 2. Then the statement x P (x) means (a) There is no integer which is the square root of 2. (b) Not every integer is the square root of 2. (c) Every integer is not the square root of 2. (d) There is an integer which is the square root of 2. (e) Don t know. 15

16 16. Let L(x, y) denote the predicate x loves y and the universe of discourse for x and y be the set of all people. Then the statement x y L(x, y) means (a) Somebody loves everybody. (b) There is someone who everybody loves. (c) Nobody loves everybody. (d) Everybody loves somebody. 16

17 17. Let L(x, y) denote the predicate x loves y and the universe of discourse for x and y be the set of all people. Then the statement x y L(x, y) means (a) Nobody loves everybody. (b) Everybody loves somebody. (c) There is someone who everybody loves. (d) Somebody loves everybody.

18 18. Poll on assignment (a) I have finished doing the assignment. (b) I have finished most of the assignment. (c) I have started the assignment but have not made much headway. (d) I have not even looked at the assignment yet. 17

19 19. The statement x y z x 2 + y > z (a) is true. (b) is false. (c) is sometimes true, sometimes false. 18

20 20. The statement x [P (x) Q(x)] [ xp (x) xq(x)] (a) is true. (b) is false. (c) is sometimes true, sometimes false. 19

21 21. Let the universe of discourse be all integers. Then the truth value of x y z x 2 + y > z is: (a) true (b) false (c) sometimes true, sometimes false 20

22 22. ( x P (x) Q(x)) x Q(x) P (x) (a) is true. (b) is false. (c) is sometimes true, sometimes false. (d) Don t know. 21

23 23. Let the universe of discourse be all students and let C(x, y) be the statement x has chatted with y. Then the statement x y C(y, x) means (a) There is a student who has not chatted with any student. (b) There is no student who has chatted with another student. (c) There is a student who has chatted with all students. (d) Every student has chatted with some student. (e) There is a student who no student has chatted with. 22

24 24. Let the universe of discourse be all students and let C(x, y) be the statement x has chatted with y. Then the statement x y z z x z y x y C(x, y) C(x, z) means (a) Every student has chatted with y and z. (b) Every student has chatted with at least two other students. (c) y and z and x are all different and x has chatted with y and z. (d) There is a student who has chatted with every student. (e) There are two students who have chatted with every student. 23

25 25. Let the universe of discourse be all students and let C(x, y) be the statement x has chatted with y. Then the statement x C(x, John) y y John C(x, y) means (a) There is a student who has chatted only with John. (b) There is a student who is the only one John has chatted with. (c) Some students have chatted only with John. (d) Every student has chatted with John. (e) If a student has chatted with John, then he hasn t chatted with anyone else. 24

26 26. Poll on background reading: I have caught up on my reading upto (a) Section 1.4 (b) Section 1.3 (c) Section 1.2 (d) Section 1.1 (e) I haven t opened the textbook yet. 25

27 27. Poll on assignment (a) I have finished doing the assignment. (b) I have finished most of the assignment. (c) I have started the assignment but have not made much headway. (d) I have not even looked at the assignment yet. 26

28 28. What rule of inference is used in the following argument? If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today. (a) Modus ponens (b) Modus tollens (c) Hypothetical syllogism (d) Simplification (e) Disjunctive syllogism 27

29 29. What rule of inference is used in the following argument? It is either hot today or the pollution is dangerous. It is not hot today. Therefore the pollution is dangerous. (a) Modus ponens (b) Modus tollens (c) Hypothetical syllogism (d) Disjunctive syllogism (e) Addition 28

30 30. In the Logic Problem, we have the following two assumptions: 1. Either logic is difficult or not many students like logic. 2. If mathematics is easy, then logic is not difficult. Is the following a valid conclusion of these assumptions? Mathematics is not easy, if many students like logic. (a) Yes (b) No (c) No clue 29

31 31. All poets are interesting people. Ann is an interesting person. Therefore, Ann is a poet. The above argument is an example of (a) a valid argument. (b) converse error (fallacy of affirming the conclusion). (c) inverse error (fallacy of denying the hypothesis). (d) circular reasoning. 30

32 32. The following argument is valid: All vegetarians eat vegetables. Some people who eat vegetables eat candy. Therefore, some vegetarians eat candy. (a) true (b) false (c) sometimes true, sometimes false 31

33 33. The following argument is valid: All movies made by John Sayles are wonderful. John Sayles made a movie about coal miners. Therefore, there is a movie about coal miners that is wonderful. (a) true (b) false (c) sometimes true, sometimes false 32

34 If x is even and y is odd, then x + 2y is even. 34. To prove the above statement using a direct proof, we start by assuming: (a) x is even, y is odd and x + 2y is odd. (b) x is even and y is odd. (c) x is even, y is odd and x + 2y is even. (d) x is odd or y is even. (e) If x is even and y is odd, then x + 2y is even. 33

35 35. Consider the statement: If 5x is even, then x is even. To prove this statement using an indirect proof, we would start by assuming that: (a) 5x is even. (b) x is even. (c) 5x is odd. (d) x is odd. (e) 5x is even and x is odd. 34

36 36. Consider the statement: If x is rational, and y is irrational, then x + y is irrational. To prove this statement using a proof by contradiction, we would start by assuming that: (a) x is rational, and y is irrational. (b) x is rational, y is irrational, and x + y is rational. (c) x is irrational, y is rational, and x + y is irrational. (d) If x is rational and y is irrational, then x + y is rational. (e) If x is irrational and y is rational, then x + y is irrational. 35

37 37. Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6}. Then the set {1, 2} is equal to (a) A B (b) A B (c) A B (d) A B (e) none of the specified sets 36

38 38. Let A = 4. Then P (A) = (a) 16 (b) 4 (c) 32 (d) 8 37

39 39. [(A B) (A B) = A] B = The above statement is (a) true (b) false (c) sometimes true, sometimes false 38

40 40. Let A and B be sets. Then {x x A x / B} can also be expressed as (a) A B (b) A B (c) A B (d) A = B (e) A / B.

41 41. Let A and B be sets. Then x x A x / B can also be expressed as (a) A B (b) A B (c) B A (d) A B (e) A / B.

42 42. Poll on test (a) I am feeling very prepared for the test. (b) I have been reviewing material, but don t yet feel completely prepared. (c) I haven t prepared for the test yet, but I think I ll be okay. (d) Help! I am really nervous. 39

43 43. Which of the following is not a function from R to R? (a) f(x) = x (b) f(x) = x (c) f(x) = 1/x (d) f(x) = x 5 40

44 44. Let f : Z Z be a function such that f(x) = (x + 2) 2. Then f is (a) one-to-one (b) not one-to-one (c) neither of the above (d) don t know 41

45 45. Let f : Z Z be a function such that f(x) = x 3 1. Then f is (a) onto (b) not onto (c) neither of the above (d) don t know 42

46 46. Let f : Z Z Z be a function such that f(m, n) = m n. Then f is (a) one-to-one and onto (b) one-to-one but not onto (c) onto but not one-to-one (d) neither one-to-one nor onto (e) don t know 43

47 47. Let f : R R be a function such that f(x) = x 3 1. Then (a) f 1 (x) = (x + 1) 1/3 (b) f 1 (x) = x 1/3 + 1 (c) f 1 (x) = x 1/3 1 (d) f is not invertible. 44

48 48. Let f and g be functions from Z to Z such that f(x) = x + 5 and g(x) = x 2 + x + 1. (a) f g(x) = x 2 + x + 6 (b) f g(x) = x x + 26 (c) f g(x) = x x + 31 (d) f g is not defined. 45

49 49. If f and f g are both onto, then does it follow that g is onto? (a) yes (b) no (c) don t know 46

50 50. For every real number x x = x The above statement is (a) false (b) true (c) sometimes true, sometimes false. 47

51 51. Poll on test: (a) It was too easy. (b) It was easy but not too easy. (c) It was reasonable, but a bit hard. (d) It was way too hard. 48

52 52. For all integers a, if a 0 then a 0. The above statement is (a) true (b) false (c) sometimes true, sometimes false 49

53 53. For all integers a, b, and c, if a b and a c, then a bc. The above statement is (a) true (b) false 50

54 54. Which of the following numbers is a prime? (a) 15 (b) 23 (c) 93 (d)

55 55. Which of the following is equal to gcd( , 2 2 7)? (a) 14 (b) 8 (c) 7 (d) 4 52

56 56. Which of the following integers is equal to 17 mod 3? (a) -2 (b) -1 (c) 1 (d) 2 53

57 57. Consider the statement If n is an even integer, then n 2 0 (mod 6) or n 2 4 (mod 6). A proof of the above statement should consider the cases: (a) n 2 = 6k; n 2 = 6k + 4 where k Z. (b) n = 2k; n = 2k + 1 where k Z. (c) n = 6k; n = 6k + 2; n = 6k + 4 where k Z. (d) n 2 = 2k; n 2 = 2k + 1 where k Z. 54

58 58. Consider the statement: If n is prime, then at least one of n + 2 and n + 4 is a composite number. To prove the above statement using a proof by contradiction, we start by assuming: (a) n, n + 2, n + 4 are all prime numbers. (b) n is a prime number and at least one of n + 2 and n + 4 is not a composite number. (c) If n is prime, then both n + 2 and n + 4 are composite numbers. (d) If n is a composite number, then n + 2 and n + 4 are both prime numbers. 55

59 59. Using the Euclidean algorithm, how many steps does it take to find gcd(280, 98)? (a) 2 (b) 3 (c) 4 (d) 6 56

60 60. Consider the statement If n is an even integer, then n 2 0 (mod 6) or n 2 4 (mod 6). A proof of the above statement should consider the cases: (a) n 2 = 6k; n 2 = 6k + 4 where k Z. (b) n = 2k; n = 2k + 1 where k Z. (c) n = 6k; n = 6k + 2; n = 6k + 4 where k Z. (d) n 2 = 2k; n 2 = 2k + 1 where k Z. 57

61 61. Using the Euclidean algorithm, find gcd(441, 28). It is (a) 2 (b) 4 (c) 7 (d) 14 58

62 62. Consider the statement n = 3(5 n+1 1)/4 whenever n is a non-negative integer. In the basis step of a proof by induction, we need to verify that (a) = 3(5 1 1)/4 (b) = 3(5 2 1)/4 (c) 3 = 3(5 1)/4 59

63 63. Let P (n) be the statement n(n + 1) = n(n + 1)(n + 2)/3 Then the statement P (n + 1) is (a) (n + 1)(n + 2) = (n)(n + 1)(n + 2)/3 (b) (n + 1)(n + 2) = (n + 1)(n + 2)(n + 3)/3 (c) (n + 2)(n + 3) = (n + 1)(n + 2)(n + 3)/3 (d) (n)(n + 2) = (n)(n + 2)(n + 3)/3 60

64 64. equals (a) 1/2 (b) 2/ n n + 1 (c) (n + 1)/(n + 2) (d) n/(n + 1) (e) None of the above 61

65 65. Consider the statement H 2 n > 1 + n/2 whenever n is a non-negative integer and H j is the j-th harmonic number. In the basis step of a proof by induction, we need to verify that (a) H 0 = (b) H 1 = 1 (c) H 0 1 (d) H 1 1 (e) H 1 3/2 62

66 66. For which non-negative integers is n 2 n!? Give the most accurate answer. (a) n 1 (b) n 2 (c) n 3 (d) n 4 (e) n 5 63

67 67. What is wrong with the following proof of the statement All horses have the same color. Let P (n) be the proposition that all horses in any set of n horses have the same color. Basis: Clearly, P (1) is true. Induction step: Assume P (k) is true, so that all horses in any set of k horses have the same color. Now consider a set of k + 1 horses, number them as 1, 2,... k + 1. Now by the inductive hypothesis, the first k of them must have the same color, and the last k must also have the same color. Therefore, they must all have the same color and P (k + 1) is true. 64

68 68. We want to prove by induction that n 2 1 is divisible by 8 whenever n is an odd positive integer. In the inductive step, we assume that 8 divides k 2 1 for some odd positive integer k. What do we need to prove? (a) 8 divides k 1. (b) 8 divides k 2 1. (c) 8 divides (k + 1) 2 1 (d) 8 divides (k + 2) 2 1 (e) Don t know 65

69 69. How many subsets does a set of n elements have? (a) n (b) n 2 (c) 2 n (d) n! (e) Don t know 66

70 70. Consider the following recursive definition for function f: f(0) = 3 f(n + 1) = 2f(n) for all n 0 Then: (a) f(3) = 24 (b) f(3) = 12 (c) f(3) = 24 (d) f(3) = 12 (e) Don t know 67

71 71. Consider the following recursive definition of a set: 0 S If i S, then i + 2 S Then S is the (a) the set of even integers. (b) the set of odd integers. (c) the set of powers of 2. (d) the set of even positive integers. (e) the set of even non-negative integers. 68

72 72. Poll on test: (a) It was too easy. (b) It was easy but not too easy. (c) It was the perfect level of difficulty. (d) It was reasonable, but a bit hard. (e) It was way too hard. 70

73 73. Consider the relation R on the set {1, 2, 3, 4, 5} where xry if and only if x > y. What is the number of non-zero entries in the matrix for R? (a) 9 (b) 10 (c) 15 (d) 25

74 74. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is reflexive. (a) true (b) false 71

75 75. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is symmetric. (a) true (b) false 72

76 76. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is anti-symmetric. (a) true (b) false 73

77 77. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is transitive. (a) true (b) false 74

78 78. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is transitive. (a) true (b) false 75

79 79. Let R Z Z be the relation defined as follows: (x, y) R if and only if x + y = 0 Then R is transitive. (a) true (b) false 76

80 80. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x = y + 1 or x = y 1 Then R is reflexive. (a) true (b) false 77

81 81. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x = y + 1 or x = y 1 Then R is symmetric. (a) true (b) false 78

82 82. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x = y + 1 or x = y 1 Then R is anti-symmetric. (a) true (b) false 79

83 83. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x = y + 1 or x = y 1 Then R is transitive. (a) true (b) false 80

84 84. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x y 2 Then R is reflexive. (a) true (b) false 81

85 85. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x y 2 Then R is symmetric. (a) true (b) false 82

86 86. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x y 2 Then R is anti-symmetric. (a) true (b) false 83

87 87. Let R be a relation on the real numbers defined as follows: (x, y) R if and only if x y 2 Then R is transitive. (a) true (b) false 84

88 88. I have filled out the online course evaluation for this course. (a) Yes (b) No 85

89 89. Let R be a relation on a set of n elements. What is the maximum number of arcs in a shortest path between two nodes in the graph of the relation? (a) n 1 (b) n (c) n 2 (d) No bound, there could be an infinite number of arcs. 86

90 90. Consider the relation R on the set of integers where xry if and only if x 2 = y 2. (a) R is an equivalence relation. (b) R is not an equivalence relation. 87

91 91. Consider the following equivalence relation R on Z Z: (a, b)r(c, d) if and only if a b = c d. Then the equivalence classes of R are: (a) A k = {(a, b) a, b Z and a b = k} for every k Z (b) A k = {(a + k, k) a Z} for every k Z (c) Both of the above (d) None of the above (e) No idea 88

92 92. Let R be a relation on a set A. Then if R is transitive and symmetric, R must also be reflexive. The above statement is (a) true (b) false 89

93 93. How many relations can be defined on a set of n elements? (a) 2 n (b) n 2 (c) 4n 2 (d) 2 n2 90

94 94. If R is reflexive, then R n must be reflexive for all n 1. (a) true (b) false 91

95 95. If R is anti-symmetric, then R 2 must be anti-symmetric. (a) true (b) false 92

96 96. Consider the relation R = {(a, b), (b, c), (a, d), (c, d), (d, d)} on the set {a, b, c, d}. How many elements does the reflexive closure of R have? (a) 8 (b) 4 (c) 5 (d) 9 93

97 97. If R is a symmetric relation, then its symmetric closure is R. The above statement is (a) true (b) false 94

98 98. Given the matrix representation of an anti-symmetric relation R, you can derive the matrix for the anti-symmetric closure of R by (a) making the entry (x, y) a 1 if both (x, y) and (y, x) were zeroes. (b) making the entry (x, y) a 0 if both (x, y) and (y, x) were ones. (c) zeroing out the diagonal of the matrix. (d) none of the above. 95