2. Power set of empty set has exactly _____ subset.
Correct : A. one
3. The set O of odd positive integers less than 10 can be expressed by ___________
Correct : B. {1, 3, 5, 7, 9}
4. What is the cardinality of the set of odd positive integers less than 10?
Correct : B. 5
5. Which of the following two sets are equal?
Correct : C. a = {1, 2, 3} and b = {2, 1, 3}
6. The set of positive integers is ________.
Correct : A. infinite
7. What is the Cardinality of the Power set of the set {0, 1, 2}.
Correct : A. 8
8. The members of the set S = {x
x is the square of an integer and x < 100} is _________________.
Correct : B. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
9. The union of the sets {1, 2, 5} and {1, 2, 6} is the set _______________.
Correct : B. {1, 2, 5, 6}
10. The intersection of the sets {1, 2, 5} and {1, 2, 6} is the set ___________.
Correct : A. {1, 2}
11. Two sets are called disjoint if there _____________ is the empty set.
Correct : C. intersection
12. Which of the following two sets are disjoint?
Correct : D. {1, 3, 5} and {2, 4, 6}
13. The difference of {1, 2, 3} and {1, 2, 5} is the set _________.
Correct : C. {3}
14. The complement of the set A is _____________.
Correct : B. u – a
15. The bit strings for the sets are 1111100000 and 1010101010. The union of these sets is ____________.
Correct : D. 1111101010
16. The set difference of the set A with null set is ________.
Correct : A. A
17. If A = {a,b,{a,c}, ∅}, then A - {a,c} is
Correct : A. {a, b, ∅}
18. The set (A - B) – C is equal to the set
Correct : D. (a ∪ b) – c
19. Among the integers 1 to 300, the number of integers which are divisible by 3 or 5 is
Correct : D. 140
20. Using Induction Principle if 13 = 1, 23 = 3 + 5, 33 = 7 + 9 + 11, then
Correct : C. 43 = 13 + 15 + 17 + 19
21. By mathematical Induction 2n> n3
Correct : D. for n ≥ 10
22. The symmetric difference A ⊕ B is the set
Correct : B. (a∪ b) – (a∩ b)
23. If A is the set of students who play crocket, B is the set of students who play football then the set of students who play either football or cricket, but not both, can be symbolically depicted as the set
Correct : A. a ⊕ b
24. Let A and B be two sets in the same universal set. Then A – B =
Correct : C. a b
25. The number of subsets of a set containing n elements is
Correct : D. 2n
26. The set O of odd positive integers less than 10 can be expressed by ___________ .
Correct : B. {1, 3, 5, 7, 9}
27. he set of positive integers is _________ .
Correct : A. infinite
28. If p ˄ q is T, then
Correct : B. p is f, q is t
29. If p →q is F, then
Correct : D. p is t, q is f
30. The statement from ∼ (p ˄ q) is logically equivalent to
Correct : A. ∼ p ˅ ∼ q
31. p → p is logically equivalent to
Correct : B. tautology
32. The converse of p → q is
Correct : D. q → p
33. Let p: Mohan is rich, q : Mohan is happy, then the statement: Mohan is rich, but Mohan is not happy in symbolic form is
Correct : D. p ˄ ∼ q
34. Let p: I will get a job, q: I pass the exam, then the statement form: I will get a job only if I pass the exam, in symbolic from is
Correct : A. p → q
35. Let p denote the statement: “Gopal is tall”, q: “Gopal is handsome”. Then the negation of the statement Gopal is tall, but not handsome,in symbolic form is:
Correct : B. ∼ p ˅ q
36. If p ˄ (p → q) is T, then
Correct : C. p is t, q is t
37. If (∼ (p ˅ q)) → q is F, then
Correct : B. p is f, q is t
38. If (∼ p → r) ˄ (p ↔ q) is T and r is F, then truth values of p and q are:
Correct : A. p is t, q is t
39. If ((p → q ) → q) → p is F, then
Correct : C. p is f, q is t
40. (p ˄ (p → q )) → q is logically equivalent to
Correct : C. tautology
41. If (p ˅ q) ˄ (~ p˅ ~q) is F, then
Correct : D. p and q must have same truth values
42. Let p denote the statement: “I finish my homework before dinner”, q: “It rains” and r: “I will go for a walk”, the representative of the following statement: if I finish my homework before dinner and it does not rain, then I will go for walk is
Correct : B. (p ˄ ~q )→ r
43. The contrapositive of p →q is
Correct : A. ~ q → ~ p
44. Which of the following is declarative statement?
Correct : B. three is divisible by 3.
45. Which of the proposition is p ^ (~p v q) is
Correct : C. logically equivalent to p ^ q
46. The relation R defined in A = {1, 2, 3} by aRb, if
a2 – b2
£ 5. Which of the following is false?
Correct : D. range of r = {5}
47. The relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(x, y) :
x2 – y2
< 16} is given by
Correct : D. none of the above
48. If R = {x, y) : x, y Î Z, x2 + y2 £ 4} is a relation in z, then domain of R is
Correct : C. {– 2, – 1, 0, 1, 2}
49. If A = { (1, 2, 3}, then the relation R = {(2, 3)} in A is
Correct : D. not transitive
50. Let X be a family of sets and R be a relation in X, defined by ‘A is disjoint from B’. Then, R is
Correct : B. symmetric
51. R is a relation defined in Z by aRb if and only if ab ³ 0, then R is
Correct : D. equivalence
52. Let a relation R in the set R of real numbers be defined as (a, b) Î R if and only if 1 + ab > 0 for all a, bÎR. The relation R is
Correct : A. reflexive and symmetric
53. If R be relation ‘<‘ from A = {1, 2, 3, 4} to B = {1, 3, 5} ie, (a, b) Î R iff a < b, then RoR– 1 is
Correct : C. {(3, 3), (3, 5), (5, 3), (5, 5)}
54. R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3. The relation R – 1 is
Correct : B. {(8, 11), (10, 13)}
55. R is a relation on N given by N = {(x, y): 4x + 3y = 20}. Which of the following belongs to R?
Correct : D. (2, 4)
56. The relation R defined on the set of natural numbers as {(a, b): a differs from b by 3} is given
Correct : B. { (4, 1), (5, 2), (6, 3), ….}
57. Two finite sets A and B have m and n elements respectively. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m is
Correct : A. 7
58. Let X and Y be the sets of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then, n (X ÇY) is equal to
Correct : D. 12
59. Let R = { ( 3, 3 ) ( 6, 6 ) ( ( 9, 9 ) ( 12, 12 ), ( 6, 12 ) ( 3, 9 ) ( 3, 12 ), ( 3, 6 ) } be a relation on the set A = { 3, 6, 9, 12 }. The relation is
Correct : A. reflexive and transitive
60. Let f : ( - 1, 1 ) → B be a function defined by f ( x ) = 2 1 x 1 2x tan - - , then f is both one-one and onto when B is the interval
Correct : D. ((-π)/2,π/2)
61. Let R be the set of real numbers. If f : R → R is a function defined by f ( x ) = x2 , then f is]
Correct : D. none of these
62. Which of the following statement is a proposition?
Correct : D. the only odd prime number is 2
63. What is the value of x after this statement, assuming the initial value of x is 5?
‘If x equals to one then x=x+2 else x=0’.
Correct : C. 0
64. Let P: I am in Bangalore.; Q: I love cricket.; then q -> p(q implies p) is?
Correct : A. if i love cricket then i am in bangalore
65. Let P: If Sahil bowls, Saurabh hits a century.; Q: If Raju bowls, Sahil gets out on first ball. Now if P is true and Q is false then which of the following can be true?
Correct : C. sahil bowled and saurabh hits a century
66. Let P: I am in Delhi.; Q: Delhi is clean.; then q ^ p(q and p) is?
Correct : A. delhi is clean and i am in delhi
67. Let P: This is a great website, Q: You should not come back here. Then ‘This is a great website and you should come back here.’ is best represented by?
Correct : B. p ∧ ~q
68. Let P: We should be honest., Q: We should be dedicated., R: We should be overconfident. Then ‘We should be honest or dedicated but not overconfident.’ is best represented by?
Correct : D. p v q ∧ ~r
69. The compound propositions p and q are called logically equivalent if is a tautology.
Correct : A. p ↔ q
70. p → q is logically equivalent to
Correct : C. ¬p ∨ q
71. p ∨ q is logically equivalent to
Correct : D. ¬p → q
72. ¬ (p ↔ q) is logically equivalent to
Correct : B. p↔¬q
73. Which of the following statement is correct?
Correct : D. all of mentioned
74. p ↔ q is logically equivalent to
Correct : C. (p → q) ∧ (q → p)
75. (p → q) ∧ (p → r) is logically equivalent to
Correct : A. p → (q ∧ r)
76. (p → r) ∨ (q → r) is logically equivalent to
Correct : C. (p ∧ q) → r
77. Let P (x) denote the statement “x >7.” Which of these have truth value true?
Correct : D. p (9)
78. The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.
Correct : D. ∀x(c(x) → f (x))
79. The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.
Correct : C. ∃x (f (x) ∧ p (x))
80. ”Everyone wants to learn cosmology.” This argument may be true for which domains?
Correct : C. both of the mentioned
81. Let domain of m includes all students, P
(m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.
Correct : D. no student spends more than 2 hours in playing polo
82. Translate ∀x∃y(x < y) in English, considering domain as a real number for both the variable.
Correct : A. for all real number x there exists a real number y such that x is less than y
83. “The product of two negative real numbers is not negative.” Is given by?
Correct : D. ∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
84. Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”
Correct : A. ∀x l(x, joy)
85. Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.
Correct : C. amit does not like south indian dishes.
86. Use quantifiers and predicates with more than one variable to express, “There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
Correct : A. ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
87. Find a counterexample of ∀x∀y(xy > y), where the domain for all variables consists of all integers.
Correct : C. both x = -1, y = 17 and x = -2 y = 8
88. Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”
Correct : B. modus ponens
89. Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”
Correct : A. modus tollens
90. Which rule of inference is used, ”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”
Correct : C. addition
91. What rules of inference are used in this argument?
“All students in this science class has taken a course in physics” and “Marry is a student in this class” imply the conclusion “Marry has taken a course in physics.”
Correct : A. universal instantiation
92. What rules of inference are used in this argument?
“It is either colder than Himalaya today or the pollution is harmful. It is hotter than Himalaya today. Therefore, the pollution is harmful.”
Correct : C. disjunctive syllogism
93. The premises (p ∧ q) ∨ r and r → s imply which of the conclusion?
Correct : B. p ∨ s
94. What rules of inference are used in this argument?
“Jay is an awesome student. Jay is also a good dancer. Therefore, Jay is an awesome student and a good dancer.”
Correct : A. conjunction
95. “Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that
Correct : D. parul is out for a trip or raju is playing chess
96. Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove
Correct : A. ∀np ((n) → q(n))
97. Which of the following can only be used in disproving the statements?
Correct : C. counter example
98. When to proof P→Q true, we proof P false, that type of proof is known as
Correct : C. vacuous proof
99. In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?
Correct : B. proof by contradiction
100. A proof covering all the possible cases, such type of proofs are known as