Discrete Mathematics: Functions and Sets - Questions and Answers

Test your knowledge of functions and sets in discrete mathematics with this comprehensive Q&A. This resource covers key concepts, definitions, and examples, providing a valuable tool for self-assessment and exam preparation.

Discrete Mathematics: Functions and Sets - Questions and Answers

This section presents questions and answers related to functions and sets in discrete mathematics.

Questions and Answers

1. A set containing an integer that is neither positive nor negative is a(n):
   1. Finite set
   2. Empty set
   3. Non-empty set
   4. Both non-empty and finite set

   Answer: (d) Both non-empty and finite set

   Explanation: The set {0} is a non-empty, finite set.

2. The set of prime numbers is a(n):
   1. Infinite set
   2. Not a set
   3. Finite set
   4. Empty set

   Answer: (a) Infinite set

   Explanation: There are infinitely many prime numbers.

3. A set containing real numbers in the range [1, 2] is a(n):
   1. Empty set
   2. Finite set
   3. Infinite set
   4. None of the mentioned

   Answer: (c) Infinite set

   Explanation: There are infinitely many real numbers between 1 and 2.

4. Which of the following is a subset of {1, 2, 3, 4}?:
   1. {1, 2}
   2. {1, 2, 3}
   3. {1}
   4. All of the mentioned

   Answer: (d) All of the mentioned

5. The set of positive prime integers that divide 72 is:
   1. {∅}
   2. {3, 5, 7}
   3. {2, 3}
   4. {2, 3, 7}

   Answer: (c) {2, 3}

   Explanation: The prime factors of 72 are 2 and 3.

6. The power set of the empty set has how many elements?:
   1. Two
   2. One
   3. Zero
   4. Three

   Answer: (b) One

   Explanation: The power set of the empty set contains only the empty set itself.

7. The Cartesian product of A = {1, 2} and B = {a, b} is:
   1. {(1, a), (1, b), (2, a), (b, b)}
   2. {(1, 1), (a, a), (2, a), (1, b)}
   3. {(1, 1), (2, 2), (a, a), (b, b)}
   4. {(1, a), (2, a), (1, b), (2, b)}

   Answer: (d) {(1, a), (2, a), (1, b), (2, b)}

8. The set of integers whose squares are less than 100 is:
   1. {0, 2, 4, 5, 9, 55, 46, 49, 99, 81}
   2. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
   3. {1, 4, 9, 16}
   4. {0, 1, 4, 9, 25, 36, 49, 123}

   Answer: (b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

9. The intersection of {1, 2, 8, 9, 10, 5} and {1, 2, 6, 10, 12} is:
   1. {5, 6, 12, 15}
   2. {1, 2, 10}
   3. {2, 5, 10, 9}
   4. {1, 6, 12, 9, 8}

   Answer: (b) {1, 2, 10}

10. The set difference {1, 2, 5, 6} - {3, 6, 8} is:
    1. {3, 8}
    2. {1, 3}
    3. {1, 2, 5}
    4. {2, 5}

    Answer: (c) {1, 2, 5}

11. If n(A) = 20, n(B) = 30, and n(A ∪ B) = 40, then n(A ∩ B) =:
    1. 30
    2. 20
    3. 40
    4. 10

    Answer: (d) 10

    Explanation: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

12. In a group of 16 players, 10 play football, 6 play only cricket. How many play only football?:
    1. 16
    2. 4
    3. 8
    4. 10

    Answer: (b) 4

    Explanation: 10 (football) - 4 (football and cricket) = 4

13. Which of the following is a discrete object?:
    1. Integers
    2. People
    3. Rational numbers
    4. All of the mentioned

    Answer: (d) All of the mentioned

14. Which pair of sets are equal?:
    1. X = {5, 6} and Y = {6}
    2. X = {5, 6, 9} and Y = {5, 6}
    3. X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}
    4. X = {5, 6} and Y = {5, 6, 3}

    Answer: (c) X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}

15. The cardinality of the power set of {1, 5, 6} is:
    1. 8
    2. 5
    3. 6
    4. 10

    Answer: (a) 8

    Explanation: The power set of a set with n elements has 2ⁿ elements.

16. Is (Y x X) equal to (X x Y)?:
    1. Yes
    2. No
    3. None of the above
    4. I don't know

    Answer: (b) No

17. How many elements are in the power set of A = {a, b, c}?:
    1. 4
    2. 6
    3. 2
    4. 5

    Answer: (a) 4

18. How many main branches of mathematics are there?:
    1. Mostly 2 types
    2. Mostly 3 types
    3. Mostly 5 types
    4. Mostly 4 types

    Answer: (a) Mostly 2 types

    Explanation: Discrete and continuous mathematics.

19. Which of these is NOT a function type in mathematics?:
    1. One-to-many
    2. Many-to-one
    3. One-to-one
    4. All of the mentioned

    Answer: (a) One-to-many

20. Which of the following is NOT a property of an injective function?:
    1. One-to-one
    2. Many-to-one
    3. Onto
    4. None of the mentioned

    Answer: (b) Many-to-one

21. How many injections are there from a set with 4 elements to a set with 5 elements?:
    1. 120
    2. 24
    3. 64
    4. 144

    Answer: (a) 120

22. If f and g are onto functions, then g∘f is a(n):
    1. Into function
    2. One-to-one function
    3. Onto function
    4. One-to-many function

    Answer: (c) Onto function

23. How many bytes are needed to encode 2000 bits?:
    1. 8
    2. 5
    3. 2
    4. 4

    Answer: (c) 250

    Explanation: 1 byte = 8 bits, so 2000 bits = 250 bytes

24. How many even positive integers are less than 20?:
    1. 8
    2. 10
    3. 9
    4. 10

    Answer: (c) 9

25. The union of X = {2, 8, 12, 15, 16} and Y = {8, 16, 15, 18} is:
    1. {2, 8, 12, 15, 16}
    2. {8, 16, 15, 18, 9}
    3. {8, 16, 15}
    4. {2, 8, 12, 15, 16, 18}

    Answer: (d) {2, 8, 12, 15, 16, 18}

26. What does the floor function do?:
    1. Rounds a real number up to the nearest integer.
    2. Rounds a real number down to the nearest integer.
    3. Rounds a real number to the nearest integer.
    4. None of the above

    Answer: (b) Rounds a real number down to the nearest integer.

27. What does the ceiling function do?:
    1. Rounds a real number down to the nearest integer.
    2. Rounds a real number up to the nearest integer.
    3. Rounds a real number to the nearest integer.
    4. All of the above are not correct

    Answer: (b) Rounds a real number up to the nearest integer.

28. What is Floor(8.4) + Ceiling(9.9)?:
    1. 18
    2. 17
    3. 20
    4. 19

    Answer: (a) 18

29. What are the maximum values of Floor(a+b) and Ceiling(a+b) if a and b are positive numbers less than 1?:
    1. Ceil(a+b) = 1 and Floor(a+b) = 0
    2. Ceil(a+b) = 1 and Floor(a+b) = 1
    3. Ceil(a+b) = 0 and Floor(a+b) = 1
    4. Ceil(a+b) = 1 and Floor(a+b) = 2

    Answer: (b) Ceil(a+b) = 1 and Floor(a+b) = 1

30. A set containing only the integer 0 is a(n):
    1. Finite set
    2. Empty set
    3. Non-empty set
    4. Both non-empty and finite set

    Answer: (d) Both non-empty and finite set

31. The set of prime numbers is a(n):
    1. Infinite set
    2. Not a set
    3. Finite set
    4. Empty set

    Answer: (a) Infinite set

32. A set containing all real numbers between 1 and 2 is a(n):
    1. Empty set
    2. Finite set
    3. Infinite set
    4. None of the mentioned

    Answer: (c) Infinite set

33. Which is a subset of {1, 2, 3, 4}?:
    1. {1, 2}
    2. {1, 2, 3}
    3. {1}
    4. All of the mentioned

    Answer: (d) All of the mentioned

34. In roster form, the set of positive prime integers that divide 72 is:
    1. {∅}
    2. {3, 5, 7}
    3. {2, 3}
    4. {2, 3, 7}

    Answer: (c) {2, 3}

35. The power set of the empty set has how many elements?:
    1. Two
    2. One
    3. Zero
    4. Three

    Answer: (b) One

36. The Cartesian product of A = {1, 2} and B = {a, b} is:
    1. {(1, a), (1, b), (2, a), (b, b)}
    2. {(1, 1), (a, a), (2, a), (1, b)}
    3. {(1, 1), (2, 2), (a, a), (b, b)}
    4. {(1, a), (2, a), (1, b), (2, b)}

    Answer: (d) {(1, a), (2, a), (1, b), (2, b)}

37. The set of integers whose squares are less than 100 is:
    1. {0, 2, 4, 5, 9, 55, 46, 49, 99, 81}
    2. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
    3. {1, 4, 9, 16}
    4. {0, 1, 4, 9, 25, 36, 49, 123}

    Answer: (b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

38. The intersection of {1, 2, 8, 9, 10, 5} and {1, 2, 6, 10, 12} is:
    1. {5, 6, 12, 15}
    2. {1, 2, 10}
    3. {2, 5, 10, 9}
    4. {1, 6, 12, 9, 8}

    Answer: (b) {1, 2, 10}

39. The set difference {1, 2, 5, 6} - {3, 6, 8} is:
    1. {3, 8}
    2. {1, 3}
    3. {1, 2, 5}
    4. {2, 5}

    Answer: (c) {1, 2, 5}

40. If n(A) = 20, n(B) = 30, and n(A∪B) = 40, then n(A∩B) is:
    1. 30
    2. 20
    3. 40
    4. 10

    Answer: (d) 10

41. If there are 16 players, 10 play football, and 6 play only cricket, how many play only football?:
    1. 16
    2. 4
    3. 8
    4. 10

    Answer: (b) 4

42. Which of these is a discrete object?:
    1. Integers
    2. People
    3. Rational numbers
    4. All of the mentioned

    Answer: (d) All of the mentioned

43. Which pair of sets are equal?:
    1. X = {5, 6} and Y = {6}
    2. X = {5, 6, 9} and Y = {5, 6}
    3. X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}
    4. X = {5, 6} and Y = {5, 6, 3}

    Answer: (c) X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}

44. The cardinality of the power set of {1, 5, 6} is:
    1. 8
    2. 5
    3. 6
    4. 10

    Answer: (a) 8

45. Is the Cartesian product (Y x X) equal to (X x Y)?:
    1. Yes
    2. No
    3. None of the above
    4. I don't know

    Answer: (b) No

46. How many elements are in the power set of A = {a, b, c}?:
    1. 4
    2. 6
    3. 2
    4. 5

    Answer: (a) 4

47. Discrete mathematics primarily deals with:
    1. Continuous quantities
    2. Discrete quantities
    3. Both continuous and discrete quantities
    4. None of the above

    Answer: (b) Discrete quantities

48. Discrete objects are characterized by being:
    1. Continuous and unbounded
    2. Distinct and countable
    3. Interconnected and overlapping
    4. None of the above

    Answer: (b) Distinct and countable

49. Which of the following is NOT a discrete set?:
    1. The set of all even numbers less than 100
    2. The set of all integers
    3. The set of all real numbers between 0 and 1
    4. The set of all prime numbers

    Answer: (c) The set of all real numbers between 0 and 1

50. If sets X and Y have 7 and 8 elements, respectively, how many relations exist between them?:
    1. 256
    2. 272
    3. 356
    4. 56

    Answer: (a) 2⁵⁶

51. On the set {0, 1, 2, 3}, how many reflexive closures are there for the relation {(0, 1), (1, 1), (1, 3), (2, 1), (2, 2), (3, 0)}?:
    1. 26
    2. 36
    3. 8
    4. 6

    Answer: (d) 6

52. What is the transitive closure of R = {(0,1), (1,2), (2,2), (3,4), (5,3), (5,4)} on the set A = {0, 1, 2, 3, 4, 5}?:
    1. {(0,0), (4,4), (5,5), (1,1), (2,2), (3,3)}
    2. {(0,1), (1,2), (2,2), (3,4)}
    3. {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}
    4. {(0,1), (5,3), (5,4), (1,1), (2,2)}

    Answer: (c) {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}

53. Which statement is FALSE if X and Y are relations on S?:
    1. Their intersection is a relation on S.
    2. Their union is a relation on S.
    3. Their difference is a relation on S.
    4. Their symmetric closure is not a relation.

    Answer: (d) Their symmetric closure is not a relation.

54. Which of these is NOT a valid law of Boolean algebra?:
    1. (A + B)(A + C) = A + (B × C)
    2. A + A = A
    3. A × B = B × A
    4. All of the above are true

    Answer: (a) (A + B)(A + C) = A + (B × C)

55. What is the dual of AND?:
    1. OR
    2. AND
    3. NOT
    4. Exclusive OR (XOR)

    Answer: (a) OR

56. Which area of mathematics is described by the set {1, 0}?:
    1. Logical algebra
    2. Boolean algebra
    3. Set Theory
    4. Matrices

    Answer: (b) Boolean algebra

57. How many distinct ways are there to express a logical proposition using only AND and OR?:
    1. 16
    2. 32
    3. 24
    4. None of the above

    Answer: (b) 32

58. Which statement is correct for a symmetric matrix A?:
    1. A = A^T
    2. A = -A^T
    3. Diagonal elements are all 1
    4. Diagonal elements are all 0

    Answer: (a) A = A^T

59. A matrix with one row and multiple columns is a:
    1. Diagonal Matrix
    2. Row Matrix
    3. Column Matrix
    4. None of the mentioned

    Answer: (b) Row Matrix

60. A matrix with multiple rows and only one column is a:
    1. Diagonal Matrix
    2. Row Matrix
    3. Column Matrix
    4. None of the mentioned

    Answer: (c) Column Matrix

61. To add two matrices, they must have:
1. The same number of rows and columns
2. The same number of columns
3. The same number of rows
4. The number of rows in the first matrix must equal the number of columns in the second matrix

Answer: (a) The same number of rows and columns

62. Is the statement A + B = B + A always true for matrix addition?:
    1. False
    2. True

    Answer: (b) True

63. Is the statement AB = BA always true for matrix multiplication?:
    1. False
    2. True

    Answer: (a) False

64. Which gate is considered a universal logic gate?:
    1. OR
    2. NOT
    3. AND
    4. NAND

    Answer: (d) NAND

65. In what year did Maurice Karnaugh publish the Karnaugh map?:
    1. 1952
    2. 1956
    3. 1953
    4. 1958

    Answer: (c) 1953

66. How many main canonical forms are there for Boolean expressions?:
    1. Mostly Two types
    2. Mostly Four types
    3. Mostly Three types
    4. Mostly Five types

    Answer: (a) Mostly Two types

67. Boolean algebra is primarily used in the design of:
1. Logic symbols
2. Digital computers
3. Circuit theory
4. None of the above

Answer: (b) Digital computers

68. Boolean algebra deals with how many discrete values?:
    1. Four
    2. Three
    3. Five
    4. Two

    Answer: (d) Two

69. Which search method checks each element sequentially?:
    1. Merge search
    2. Sequential search
    3. Binary search
    4. None of the mentioned

    Answer: (b) Sequential search

70. In insertion sort, which element starts the sorting process?:
    1. First
    2. Second
    3. Third
    4. Fourth

    Answer: (b) Second

71. What is the time complexity of bubble sort?:
    1. O(n)
    2. O(log n)
    3. O(n log n)
    4. O(n²)

    Answer: (d) O(n²)

72. Which algorithm uses prior outputs to calculate new outputs?:
    1. Divide and Conquer
    2. Dynamic Programming
    3. Brute Force
    4. None of them

    Answer: (b) Dynamic Programming

73. Which of the following is NOT a way to represent an algorithm?:
    1. Flowcharts
    2. Pseudocode
    3. Natural Language
    4. All of the above are valid representations

    Answer: (d) All of the above are valid representations

74. Complexity theory does NOT typically consider which case?:
    1. Average case
    2. Best case
    3. Null case
    4. Worst case

    Answer: (c) Null case

75. Discrete mathematics primarily deals with:
1. Continuous quantities
2. Discrete quantities
3. Both continuous and discrete quantities
4. None of the above

Answer: (b) Discrete quantities

76. The statement "Discrete items are distinct and isolated from one another" is:
1. TRUE
2. FALSE
3. MAYBE
4. CAN'T SAY

Answer: (a) TRUE

77. Which of the following is NOT a discrete set?:
1. The set of all even numbers less than 100
2. The set of all integers
3. The set of all real numbers between 0 and 1
4. The set of all prime numbers

Answer: (c) The set of all real numbers between 0 and 1

78. What kind of set is {0}?:
    1. Finite set
    2. Empty set
    3. Non-empty set
    4. Both non-empty and finite set

    Answer: (d) Both non-empty and finite set

79. The set of prime numbers is:
1. Infinite
2. Not a set
3. Finite
4. Empty

Answer: (a) Infinite

80. A set containing all real numbers between 1 and 2 is a(n):
    1. Empty set
    2. Finite set
    3. Infinite set
    4. None of the mentioned

    Answer: (c) Infinite set

81. Which is a subset of {1, 2, 3, 4}?:
    1. {1, 2}
    2. {1, 2, 3}
    3. {1}
    4. All of the mentioned

    Answer: (d) All of the mentioned

82. The set of prime numbers that divide 72 is:
    1. {∅}
    2. {3, 5, 7}
    3. {2, 3}
    4. {2, 3, 7}

    Answer: (c) {2, 3}

83. How many elements are in the power set of the empty set?:
    1. Two
    2. One
    3. Zero
    4. Three

    Answer: (b) One

84. What is the Cartesian product of A = {1, 2} and B = {a, b}?:
    1. {(1, a), (1, b), (2, a), (b, b)}
    2. {(1, 1), (a, a), (2, a), (1, b)}
    3. {(1, 1), (2, 2), (a, a), (b, b)}
    4. {(1, a), (2, a), (1, b), (2, b)}

    Answer: (d) {(1, a), (2, a), (1, b), (2, b)}

85. What is the set of integers whose squares are less than 100?:
    1. {0, 2, 4, 5, 9, 55, 46, 49, 99, 81}
    2. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
    3. {1, 4, 9, 16}
    4. {0, 1, 4, 9, 25, 36, 49, 123}

    Answer: (b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

86. What is the intersection of {1, 2, 8, 9, 10, 5} and {1, 2, 6, 10, 12}?:
    1. {5, 6, 12, 15}
    2. {1, 2, 10}
    3. {2, 5, 10, 9}
    4. {1, 6, 12, 9, 8}

    Answer: (b) {1, 2, 10}

87. What is the set difference {1, 2, 5, 6} - {3, 6, 8}?:
    1. {3, 8}
    2. {1, 3}
    3. {1, 2, 5}
    4. {2, 5}

    Answer: (c) {1, 2, 5}

88. If n(A) = 20, n(B) = 30, and n(A∪B) = 40, then n(A∩B) =:
1. 10
2. 20
3. 30
4. 40

Answer: (a) 10

89. If there are 16 players, 10 play football, and 6 play only cricket, how many play only football?:
    1. 16
    2. 4
    3. 8
    4. 10

    Answer: (b) 4

90. Which of these is a discrete object?:
    1. Integers
    2. People
    3. Rational numbers
    4. All of the mentioned

    Answer: (d) All of the mentioned

91. Which pair of sets are equal?:
    1. X = {5, 6} and Y = {6}
    2. X = {5, 6, 9} and Y = {5, 6}
    3. X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}
    4. X = {5, 6} and Y = {5, 6, 3}

    Answer: (c) X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}

92. The cardinality of the power set of {1, 5, 6} is:
1. 8
2. 5
3. 6
4. 10

Answer: (a) 8

93. Is (Y x X) equal to (X x Y)?:
    1. Yes
    2. No
    3. None of the above
    4. I don't know

    Answer: (b) No

94. How many elements are in the power set of A = {a, b, c}?:
    1. 4
    2. 6
    3. 2
    4. 5

    Answer: (a) 4

95. The number of ways to partition a set with 3 elements is given by:
1. 3
2. 5
3. 4
4. 6

Answer: (c) 5

96. If sets X and Y have 7 and 8 elements respectively, how many relations exist between them?:
    1. 256
    2. 272
    3. 356
    4. 56

    Answer: (a) 2⁵⁶

97. The number of reflexive closures for the relation {(0, 1), (1, 1), (1, 3), (2, 1), (2, 2), (3, 0)} on the set {0, 1, 2, 3} is:
1. 6
2. 8
3. 26
4. 36

Answer: (a) 6

98. What is the transitive closure of the relation {(0,1), (1,2), (2,2), (3,4), (5,3), (5,4)} on {0, 1, 2, 3, 4, 5}?:
    1. {(0,0), (4,4), (5,5), (1,1), (2,2), (3,3)}
    2. {(0,1), (1,2), (2,2), (3,4)}
    3. {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}
    4. {(0,1), (5,3), (5,4), (1,1), (2,2)}

    Answer: (c) {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}

99. Which statement is FALSE if X and Y are relations on S?:
    1. Their intersection is a relation on S.
    2. Their union is a relation on S.
    3. Their difference is a relation on S.
    4. Their symmetric closure is not a relation.

    Answer: (d) Their symmetric closure is not a relation.

100. Which is NOT a valid law of Boolean algebra?:
     1. (A + B)(A + C) = A + (B × C)
     2. A + A = A
     3. A × B = B × A
     4. All of the above are true

     Answer: (a) (A + B)(A + C) = A + (B × C)

101. The dual of the AND operation is:
1. OR
2. AND
3. NOT
4. XOR

Answer: (a) OR

102. The set {1, 0} is primarily associated with:
1. Logical algebra
2. Boolean algebra
3. Set Theory
4. Matrices

Answer: (b) Boolean algebra

103. How many distinct ways are there to express a logical proposition using only AND and OR (considering order and parentheses)?:
     1. 16
     2. 32
     3. 24
     4. None of the above

     Answer: (b) 32

104. A symmetric matrix A satisfies:
1. A = A^T
2. A = -A^T
3. Diagonal elements are all 1
4. Diagonal elements are all 0

Answer: (a) A = A^T

105. A matrix with one row and multiple columns is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (a) Row Matrix

106. A matrix with multiple rows and only one column is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (b) Column Matrix

107. To add two matrices, they must have:
1. The same number of rows and columns
2. The same number of columns
3. The same number of rows
4. The number of rows in the first must equal the number of columns in the second

Answer: (a) The same number of rows and columns

108. For matrix addition, A + B = B + A is:
1. False
2. True

Answer: (b) True

109. For matrix multiplication, AB = BA is:
1. False
2. True

Answer: (a) False

110. The NAND gate is considered a universal gate because:
1. It can perform all logical operations
2. It is the simplest gate
3. It is used in all computers
4. It is the fastest gate

Answer: (a) It can perform all logical operations

111. Maurice Karnaugh published the Karnaugh map in:
1. 1953
2. 1952
3. 1956
4. 1958

Answer: (a) 1953

112. How many main canonical forms exist for Boolean expressions?:
1. Two
2. Four
3. Three
4. Five

Answer: (a) Two

113. Boolean algebra is fundamentally used in the design of:
1. Logic symbols
2. Digital computers
3. Circuit theory
4. None of the above

Answer: (b) Digital computers

114. Boolean algebra operates on how many discrete values?:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two

115. A sequential search algorithm is characterized by:
1. Comparing each element sequentially
2. Using a divide-and-conquer approach
3. Sorting the list beforehand
4. None of the above

Answer: (a) Comparing each element sequentially

116. In insertion sort, the sorting process begins with which element?:
1. The first element
2. The second element
3. The middle element
4. The last element

Answer: (b) The second element

117. The worst-case time complexity of a linear search is:
1. O(log n)
2. O(n)
3. O(n log n)
4. O(n²)

Answer: (b) O(n)

118. What is the time complexity of the bubble sort algorithm?:
1. O(n)
2. O(log n)
3. O(n log n)
4. O(n²)

Answer: (d) O(n²)

119. Dynamic programming algorithms are characterized by:
1. Reusing solutions to subproblems
2. Dividing the problem into independent subproblems
3. Trying all possible solutions
4. None of the above

Answer: (a) Reusing solutions to subproblems

120. Which of the following is NOT typically considered in complexity theory?:
     1. Average case
     2. Best case
     3. Null case
     4. Worst case

     Answer: (c) Null case

121. Is the following statement true? “A field of mathematics that utilizes discrete elements is known as discrete mathematics.”:
1. True
2. False

Answer: (a) True

122. The statement "Discrete items are distinct from and isolated from one another" is:
1. True
2. False

Answer: (a) True

123. Which of the following is NOT a discrete set?:
1. The set of all even numbers less than 100
2. The set of all integers
3. The set of all real numbers between 0 and 1
4. The set of all prime numbers

Answer: (c) The set of all real numbers between 0 and 1

124. What kind of set is {0}?:
     1. Finite set
     2. Empty set
     3. Non-empty set
     4. Both non-empty and finite set

     Answer: (d) Both non-empty and finite set

125. The set of prime numbers is:
1. Infinite
2. Not a set
3. Finite
4. Empty

Answer: (a) Infinite

126. The set of real numbers between 1 and 2 is:
1. Empty
2. Finite
3. Infinite
4. None of the mentioned

Answer: (c) Infinite

127. Which is a subset of {1, 2, 3, 4}?:
     1. {1, 2}
     2. {1, 2, 3}
     3. {1}
     4. All of the mentioned

     Answer: (d) All of the mentioned

128. The set of prime factors of 72 is:
     1. {∅}
     2. {3, 5, 7}
     3. {2, 3}
     4. {2, 3, 7}

     Answer: (c) {2, 3}

129. How many elements are in the power set of the empty set?:
     1. Two
     2. One
     3. Zero
     4. Three

     Answer: (b) One

130. What is the Cartesian product of A = {1, 2} and B = {a, b}?:
     1. {(1, a), (1, b), (2, a), (b, b)}
     2. {(1, 1), (a, a), (2, a), (1, b)}
     3. {(1, 1), (2, 2), (a, a), (b, b)}
     4. {(1, a), (2, a), (1, b), (2, b)}

     Answer: (d) {(1, a), (2, a), (1, b), (2, b)}

131. The set of integers whose squares are less than 100 is:
     1. {0, 2, 4, 5, 9, 55, 46, 49, 99, 81}
     2. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
     3. {1, 4, 9, 16}
     4. {0, 1, 4, 9, 25, 36, 49, 123}

     Answer: (b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}

132. What is the intersection of {1, 2, 8, 9, 10, 5} and {1, 2, 6, 10, 12}?:
     1. {5, 6, 12, 15}
     2. {1, 2, 10}
     3. {2, 5, 10, 9}
     4. {1, 6, 12, 9, 8}

     Answer: (b) {1, 2, 10}

133. What is the set difference {1, 2, 5, 6} - {3, 6, 8}?:
     1. {3, 8}
     2. {1, 3}
     3. {1, 2, 5}
     4. {2, 5}

     Answer: (c) {1, 2, 5}

134. If n(A) = 20, n(B) = 30, n(A∪B) = 40, then n(A∩B) =:
1. 10
2. 20
3. 30
4. 40

Answer: (a) 10

135. In a group of 16 players, 10 play football, and 6 play only cricket. How many play only football?:
     1. 16
     2. 4
     3. 8
     4. 10

     Answer: (b) 4

136. Which of these is a discrete object?:
     1. Integers
     2. People
     3. Rational numbers
     4. All of the mentioned

     Answer: (d) All of the mentioned

137. Which pair of sets are equal?:
     1. X = {5, 6} and Y = {6}
     2. X = {5, 6, 9} and Y = {5, 6}
     3. X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}
     4. X = {5, 6} and Y = {5, 6, 3}

     Answer: (c) X = {5, 6, 8, 9} and Y = {6, 8, 5, 9}

138. The cardinality of the power set of {1, 5, 6} is:
1. 8
2. 5
3. 6
4. 10

Answer: (a) 8

139. Is (Y x X) equal to (X x Y)?:
     1. Yes
     2. No
     3. None of the above
     4. I don't know

     Answer: (b) No

140. How many elements are in the power set of A = {a, b, c}?:
     1. 4
     2. 6
     3. 2
     4. 5

     Answer: (a) 4

141. The number of ways to partition a set with 3 elements is:
1. 3
2. 5
3. 4
4. 6

Answer: (b) 5

142. If sets X and Y have 7 and 8 elements, the number of relations between them is:
1. 2⁵⁶
2. 256
3. 272
4. 356

Answer: (a) 2⁵⁶

143. The number of reflexive closures for the relation {(0, 1), (1, 1), (1, 3), (2, 1), (2, 2), (3, 0)} on {0, 1, 2, 3} is:
1. 6
2. 8
3. 26
4. 36

Answer: (a) 6

144. The transitive closure of {(0,1), (1,2), (2,2), (3,4), (5,3), (5,4)} on {0, 1, 2, 3, 4, 5} is:
1. {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}
2. {(0,0), (4,4), (5,5), (1,1), (2,2), (3,3)}
3. {(0,1), (1,2), (2,2), (3,4)}
4. {(0,1), (5,3), (5,4), (1,1), (2,2)}

Answer: (a) {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}

145. If X and Y are relations on S, which statement is FALSE?:
     1. Their intersection is a relation on S.
     2. Their union is a relation on S.
     3. Their difference is a relation on S.
     4. Their symmetric closure is not a relation.

     Answer: (d) Their symmetric closure is not a relation.

146. Which is NOT a valid law of Boolean algebra?:
     1. (A + B)(A + C) = A + (B × C)
     2. A + A = A
     3. A × B = B × A
     4. All of the above are true

     Answer: (a) (A + B)(A + C) = A + (B × C)

147. The dual of the AND operation is:
1. OR
2. AND
3. NOT
4. XOR

Answer: (a) OR

148. The set {1, 0} is primarily associated with:
1. Logical algebra
2. Boolean algebra
3. Set Theory
4. Matrices

Answer: (b) Boolean algebra

149. How many distinct ways are there to express a logical proposition using only AND and OR (considering order and parentheses)?:
     1. 16
     2. 32
     3. 24
     4. None of the above

     Answer: (b) 32

150. A symmetric matrix A satisfies:
1. A = A^T
2. A = -A^T
3. Diagonal elements are all 1
4. Diagonal elements are all 0

Answer: (a) A = A^T

151. A matrix with one row and multiple columns is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (a) Row Matrix

152. A matrix with multiple rows and only one column is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (b) Column Matrix

153. For matrix addition, the matrices must have:
1. The same number of rows and columns
2. The same number of columns
3. The same number of rows
4. The number of rows in the first equals the number of columns in the second

Answer: (a) The same number of rows and columns

154. The statement A + B = B + A is true for matrix addition:
1. True
2. False

Answer: (a) True

155. The statement AB = BA is true for matrix multiplication:
1. True
2. False

Answer: (b) False

156. A universal logic gate is:
1. NAND
2. OR
3. NOT
4. AND

Answer: (a) NAND

157. Maurice Karnaugh published his map in:
1. 1953
2. 1952
3. 1956
4. 1958

Answer: (a) 1953

158. The number of main canonical forms for Boolean expressions is:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two

159. Boolean algebra is mainly used in designing:
1. Logic symbols
2. Digital computers
3. Circuit theory
4. None of the above

Answer: (b) Digital computers

160. Boolean algebra deals with how many values?:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two

161. A sequential search algorithm involves:
1. Comparing each element sequentially
2. Dividing the problem into subproblems
3. Sorting the data first
4. None of the above

Answer: (a) Comparing each element sequentially

162. In insertion sort, sorting starts with the:
1. First element
2. Second element
3. Middle element
4. Last element

Answer: (b) Second element

163. The worst-case time complexity of a linear search is:
1. O(log n)
2. O(n)
3. O(n log n)
4. O(n²)

Answer: (b) O(n)

164. The time complexity of bubble sort is:
1. O(n)
2. O(log n)
3. O(n log n)
4. O(n²)

Answer: (d) O(n²)

165. Dynamic programming is characterized by:
1. Reusing solutions to subproblems
2. Dividing problems into independent subproblems
3. Trying all possible solutions
4. None of the above

Answer: (a) Reusing solutions to subproblems

166. Complexity theory does NOT typically consider the ______ case:
1. Null case
2. Average case
3. Best case
4. Worst case

Answer: (a) Null case

167. Is this statement true or false? "Real numbers, including rational and irrational numbers, are discrete":
1. True
2. False

Answer: (b) False

168. How many broad categories are there in mathematics?:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two (Discrete and Continuous)

169. Which type of mathematics deals with distinct, separate values?:
1. Discrete Mathematics
2. Continuous Mathematics
3. Non-Discrete Mathematics
4. Non-Continuous Mathematics

Answer: (a) Discrete Mathematics

170. If sets X and Y have 7 and 8 elements, respectively, how many relations exist between them?:
     1. 256
     2. 272
     3. 356
     4. 56

     Answer: (a) 2⁵⁶

     Explanation: There are 2^mn possible relations between two sets with cardinalities m and n.

171. On the set {0, 1, 2, 3}, how many reflexive closures are there for the relation {(0, 1), (1, 1), (1, 3), (2, 1), (2, 2), (3, 0)}?:
     1. 26
     2. 36
     3. 8
     4. 6

     Answer: (d) 6

172. What is the transitive closure of the relation {(0,1), (1,2), (2,2), (3,4), (5,3), (5,4)} on the set {0, 1, 2, 3, 4, 5}?:
     1. {(0,0), (4,4), (5,5), (1,1), (2,2), (3,3)}
     2. {(0,1), (1,2), (2,2), (3,4)}
     3. {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}
     4. {(0,1), (5,3), (5,4), (1,1), (2,2)}

     Answer: (c) {(0,1), (0,2), (1,2), (2,2), (3,4), (5,3), (5,4)}

173. If X and Y are relations on set S, which statement is FALSE?:
     1. Their intersection is a relation on S.
     2. Their union is a relation on S.
     3. Their difference is a relation on S.
     4. Their symmetric closure is not a relation.

     Answer: (d) Their symmetric closure is not a relation.

174. Which is NOT a valid law of Boolean algebra?:
     1. (A + B)(A + C) = A + (B × C)
     2. A + A = A
     3. A × B = B × A
     4. All of the above are true

     Answer: (a) (A + B)(A + C) = A + (B × C)

175. The dual of the AND operation is:
1. OR
2. AND
3. NOT
4. XOR

Answer: (a) OR

176. The set {1, 0} is primarily associated with:
1. Logical algebra
2. Boolean algebra
3. Set Theory
4. Matrices

Answer: (b) Boolean algebra

177. How many distinct ways are there to express a logical proposition using only AND and OR (considering order and parentheses)?:
     1. 16
     2. 32
     3. 24
     4. None of the above

     Answer: (b) 32

178. A symmetric matrix A satisfies:
1. A = A^T
2. A = -A^T
3. Diagonal elements are all 1
4. Diagonal elements are all 0

Answer: (a) A = A^T

179. A matrix with one row and multiple columns is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (a) Row Matrix

180. A matrix with multiple rows and only one column is a:
1. Row Matrix
2. Column Matrix
3. Diagonal Matrix
4. None of the mentioned

Answer: (b) Column Matrix

181. For matrix addition, the matrices must have:
1. The same number of rows and columns
2. The same number of columns
3. The same number of rows
4. The number of rows in the first equals the number of columns in the second

Answer: (a) The same number of rows and columns

182. The statement A + B = B + A is true for matrix addition:
1. True
2. False

Answer: (a) True

183. The statement AB = BA is true for matrix multiplication:
1. True
2. False

Answer: (b) False

184. A universal logic gate is:
1. NAND
2. OR
3. NOT
4. AND

Answer: (a) NAND

185. Maurice Karnaugh published his map in:
1. 1953
2. 1952
3. 1956
4. 1958

Answer: (a) 1953

186. The number of main canonical forms for Boolean expressions is:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two

187. Boolean algebra is mainly used in designing:
1. Logic symbols
2. Digital computers
3. Circuit theory
4. None of the above

Answer: (b) Digital computers

188. Boolean algebra operates on how many values?:
1. Two
2. Three
3. Four
4. Five

Answer: (a) Two

189. In a linear graph, vertices are arranged in a:
1. Line
2. Circle
3. Tree
4. None of the above

Answer: (a) Line

190. A star tree is characterized by:
1. n leaves and one internal vertex
2. n vertices and n-1 cycles
3. n vertices in a straight line
4. 0 or more connected subtrees

Answer: (a) n leaves and one internal vertex

191. A tree with exactly one center vertex is called a:
1. Central tree
2. Bicenter tree
3. Rooted tree
4. Labeled tree

Answer: (a) Central tree

192. Probability theory started to formally develop around:
1. 1654
2. 1638
3. 1674
4. 1666

Answer: (a) 1654

193. In an experiment, the set of all possible outcomes is called the:
1. Sample Space
2. Event
3. Random Experiment
4. Tossing Space

Answer: (a) Sample Space

194. How many outcomes are there in a coin toss?:
1. Two
2. One
3. Three
4. Four

Answer: (a) Two

195. The probability of getting a specific number on a fair six-sided die roll is:
1. (1/6)
2. (5/6)
3. (2/3)
4. (1/3)

Answer: (a) (1/6)

196. What is the probability of drawing an ace from a standard deck of 52 cards?:
1. (1/13)
2. (1/52)
3. (3/52)
4. (1/26)

Answer: (a) (1/13)

197. What is the probability of drawing a diamond from a standard deck of 52 cards?:
1. (1/4)
2. (1/6)
3. (3/26)
4. (1/13)

Answer: (a) (1/4)

198. Discrete probability distributions are defined by:
1. Discrete variables
2. Probability function
3. Data
4. Machine

Answer: (c) Data

199. A bag contains 5 red balls and twice as many blue balls. What is the probability of drawing a blue ball?:
1. 10
2. 20
3. 5
4. 15

Answer: (a) 2/3

200. A box contains cards numbered 2 to 101. What is the probability that a randomly selected card has a number that is a perfect square?:
1. (1/10)
2. (1/5)
3. (1/25)
4. (1/20)

Answer: (a) (1/10)

201. The mean annual salary is Rs. 48,000 with a standard deviation of Rs. 1500. What is the approximate probability that an employee earns between Rs. 45,000 and Rs. 52,000?:
1. 0.421
2. 0.42
3. 0.422
4. 0.423

Answer: (a) 0.421 (approximately)