1. Who is considered the founder of the modern theory of sets?

A) Richard Dedekind
B) Georg Cantor
C) Gottlob Frege
D) Bertrand Russell
Answer: B) Georg Cantor

2. Which of the following is an example of a well-defined collection?

A) A pack of cards
B) Five most renowned mathematicians
C) A team of the best cricket players in the world
D) Most dangerous animals
Answer: A) A pack of cards

3. The set {x : x is a natural number and 3 < x < 10} can be represented in roster form as:

A) {3, 4, 5, 6, 7, 8, 9}
B) {4, 5, 6, 7, 8, 9}
C) {3, 4, 5, 6, 7}
D) {4, 5, 6, 7}
Answer: B) {4, 5, 6, 7, 8, 9}

4. Which of the following sets is an empty set?

A) {x : x is a natural number and 1 < x < 2}
B) {x : x is an even prime number greater than 2}
C) {x : x is an odd number that satisfies x² = 4}
D) All of the above
Answer: D) All of the above

5. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B is:

A) {1, 2}
B) {3, 4}
C) {1, 5}
D) {5, 6}
Answer: B) {3, 4}

6. What does the symbol "∅" represent in set theory?

A) A universal set
B) An infinite set
C) An empty set
D) A subset
Answer: C) An empty set

7. Which of the following statements is true for finite sets?

A) They always have at least one element.
B) The number of elements is countable and finite.
C) All finite sets are subsets of infinite sets.
D) They always include zero as an element.
Answer: B) The number of elements is countable and finite.

8. If U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, and B = {2, 4, 6}, then A ∪ B is:

A) {1, 2, 3, 4, 5, 6}
B) {1, 3, 5}
C) {2, 4, 6}
D) {1, 3, 5, 6}
Answer: A) {1, 2, 3, 4, 5, 6}

9. What does De Morgan's law state about the complement of the union of two sets?

A) It equals the union of their complements.
B) It equals the intersection of their complements.
C) It is equal to the complement of their intersection.
D) It equals the universal set.
Answer: B) It equals the intersection of their complements.

10. In set theory, the interval (a, b) is defined as:

A) The set of numbers a and b only
B) The set of all real numbers between a and b, including a and b
C) The set of all real numbers between a and b, excluding a and b
D) None of the above
Answer: C) The set of all real numbers between a and b, excluding a and b

11. In roster form, the set of all vowels in the English alphabet is represented as:

A) {a, b, c, d, e}
B) {a, e, i, o, u}
C) {vowels in English alphabet}
D) {x : x is a vowel}
Answer: B) {a, e, i, o, u}

12. The symbol "∈" means:

A) Is not an element of
B) Is an element of
C) Belongs to the subset
D) Is a subset of
Answer: B) Is an element of

13. The set {x : x is an even positive integer less than 10} in roster form is:

A) {2, 4, 6, 8, 10}
B) {2, 4, 6, 8}
C) {2, 4, 8, 10}
D) {2, 3, 6, 8}
Answer: B) {2, 4, 6, 8}

14. If A = {a, b, c} and B = {b, c, d}, then A ∩ B is:

A) {a}
B) {b, c}
C) {a, b, c, d}
D) {a, d}
Answer: B) {b, c}

15. Which of the following sets is infinite?

A) The set of even numbers less than 20
B) The set of letters in the word "MATHEMATICS"
C) The set of natural numbers
D) The set of months in a year
Answer: C) The set of natural numbers

16. If A = {1, 2, 3} and B = {4, 5, 6}, then A ∪ B is:

A) {1, 2, 3, 4, 5, 6}
B) {1, 2, 3}
C) {4, 5, 6}
D) φ
Answer: A) {1, 2, 3, 4, 5, 6}

17. Which of the following describes a null set?

A) A set containing one element
B) A set with no elements
C) A set containing infinite elements
D) A set with all elements of the universal set
Answer: B) A set with no elements

18. The complement of a set A in a universal set U is denoted as:

A) A ∪ U
B) A ∩ U
C) A′
D) A - U
Answer: C) A′

19. A singleton set contains:

A) No elements
B) One element
C) Two elements
D) Infinite elements
Answer: B) One element

20. Which of the following is not a subset of {1, 2, 3}?

A) {1}
B) {1, 2}
C) {1, 2, 3, 4}
D) φ
Answer: C) {1, 2, 3, 4}

21. If A ∩ B = φ, the sets A and B are called:

A) Finite
B) Equal
C) Disjoint
D) Subsets
Answer: C) Disjoint

22. The union of two disjoint sets is:

A) An empty set
B) The universal set
C) A set containing all elements of both sets
D) None of these
Answer: C) A set containing all elements of both sets

23. The set of prime factors of 30 is:

A) {1, 2, 3, 5, 6}
B) {2, 3, 5}
C) {2, 5}
D) {3, 5}
Answer: B) {2, 3, 5}

24. Which of the following is an example of a finite set?

A) The set of all integers
B) The set of all stars in the sky
C) The set of even numbers less than 20
D) The set of all rational numbers
Answer: C) The set of even numbers less than 20

25. The equation x² - 9 = 0 has a solution set of:

A) {3}
B) {3, -3}
C) {9, -9}
D) {-3}
Answer: B) {3, -3}

26. A set that contains all possible elements under consideration is called:

A) A universal set
B) A finite set
C) An infinite set
D) An empty set
Answer: A) A universal set

27. Which set contains all elements common to A and B?

A) A ∩ B
B) A ∪ B
C) A - B
D) B - A
Answer: A) A ∩ B

28. For sets A and B, (A ∪ B)′ =:

A) A′ ∪ B′
B) A′ ∩ B′
C) A ∩ B
D) U
Answer: B) A′ ∩ B′

29. Which law states that A ∪ φ = A?

A) Distributive law
B) Identity law
C) Commutative law
D) Associative law
Answer: B) Identity law

30. Which of the following is an infinite set?

A) {1, 2, 3, 4, 5}
B) {x : x is a natural number}
C) {x : x is a vowel in the English alphabet}
D) {a, e, i, o, u}
Answer: B) {x : x is a natural number}

31. What is the complement of the set {1, 2, 3} in the universal set U = {1, 2, 3, 4, 5}?

A) {1, 2, 3, 4, 5}
B) {4, 5}
C) {2, 3, 4, 5}
D) φ
Answer: B) {4, 5}

32. Which of the following is an example of an equal set?

A) A = {1, 2, 3}, B = {3, 2, 1}
B) A = {1, 2}, B = {1, 2, 3}
C) A = {1, 3, 5}, B = {2, 4, 6}
D) A = {1, 2, 3}, B = {1, 2, 4}
Answer: A) A = {1, 2, 3}, B = {3, 2, 1}

33. The set {x : x is a letter in the word SCHOOL} in roster form is:

A) {S, C, H, O, L}
B) {S, C, H, O, L, O}
C) {H, S, O, L, C}
D) Both A and C
Answer: D) Both A and C

34. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A - B is:

A) {1, 2, 5, 6}
B) {1, 2}
C) {3, 4}
D) {5, 6}
Answer: B) {1, 2}

35. If A = {x : x is a natural number and x² < 25}, then the roster form of A is:

A) {1, 2, 3, 4, 5}
B) {1, 2, 3, 4}
C) {1, 4, 9, 16, 25}
D) {1, 4, 9, 16}
Answer: B) {1, 2, 3, 4}

36. The set of all integers is denoted by:

A) N
B) Z
C) R
D) Q
Answer: B) Z

37. If A and B are two sets, then (A ∩ B)′ is equal to:

A) A′ ∩ B′
B) A′ ∪ B′
C) (A ∪ B)′
D) φ
Answer: B) A′ ∪ B′

38. In a Venn diagram, the universal set is usually represented by:

A) A circle
B) A rectangle
C) A square
D) An ellipse
Answer: B) A rectangle

39. The union of A = {a, b, c} and B = {b, c, d} is:

A) {a, b, c, d}
B) {b, c}
C) {a, d}
D) φ
Answer: A) {a, b, c, d}

40. A subset of a universal set U contains:

A) All elements of U
B) Some or all elements of U
C) No elements of U
D) Only one element of U
Answer: B) Some or all elements of U

41. Which of the following intervals includes its endpoints?

A) (a, b)
B) [a, b]
C) (a, b]
D) {x : x > a and x < b}
Answer: B) [a, b]

42. The set {a, e, i, o, u} is a subset of:

A) {a, b, c, d, e, f, g}
B) {a, e, i, o, u, x, y, z}
C) φ
D) {e, o}
Answer: B) {a, e, i, o, u, x, y, z}

43. What is the intersection of A = {1, 2, 3} and B = {3, 4, 5}?

A) {3}
B) {1, 2, 3}
C) {3, 4}
D) φ
Answer: A) {3}

44. The difference A - B for A = {1, 3, 5, 7} and B = {3, 5} is:

A) {1, 7}
B) {3, 5}
C) {1, 3, 5, 7}
D) φ
Answer: A) {1, 7}

45. If A ⊂ B, then A ∩ B is equal to:

A) A
B) B
C) φ
D) A ∪ B
Answer: A) A

46. The universal set for the set of all triangles is:

A) The set of all quadrilaterals
B) The set of all polygons
C) The set of all geometric figures
D) The set of all shapes
Answer: C) The set of all geometric figures

47. The complement of the set of all even numbers in the set of natural numbers is:

A) The set of odd numbers
B) φ
C) The set of all integers
D) The universal set
Answer: A) The set of odd numbers

48. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then the union of A and B is:

A) {1, 2, 3, 4, 5, 6}
B) {3, 4}
C) {1, 2}
D) {5, 6}
Answer: A) {1, 2, 3, 4, 5, 6}

49. The set of all rational numbers is denoted by:

A) R
B) N
C) Q
D) Z
Answer: C) Q

50. If the universal set is U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, then A′ is:

A) {1, 3, 5}
B) {2, 4}
C) φ
D) {1, 2, 3, 4, 5}
Answer: B) {2, 4}

51. The set-builder form of the set {1, 4, 9, 16} is:

A) {x : x is a perfect square less than 20}
B) {x : x is a square of a natural number less than 5}
C) {x : x = n², where n ∈ N and n ≤ 4}
D) All of the above
Answer: D) All of the above

52. If A = {2, 4, 6} and B = {4, 6, 8}, then A ∪ B is:

A) {2, 4, 6}
B) {4, 6}
C) {2, 4, 6, 8}
D) φ
Answer: C) {2, 4, 6, 8}

53. If A = {x : x is an even integer} and B = {x : x is an odd integer}, then A ∩ B is:

A) φ
B) {0}
C) {1}
D) {x : x is a positive integer}
Answer: A) φ

54. If the universal set is U = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3}, then A′ is:

A) {4, 5, 6}
B) {1, 2, 3}
C) φ
D) {1, 2, 3, 4, 5, 6}
Answer: A) {4, 5, 6}

55. Which of the following is true for two sets A and B if A ⊂ B?

A) A - B = A
B) A ∪ B = A
C) A ∩ B = A
D) A ⊂ φ
Answer: C) A ∩ B = A

56. The roster form of the set {x : x is a positive integer less than 6} is:

A) {1, 2, 3, 4, 5}
B) {1, 2, 3, 4, 5, 6}
C) {2, 3, 4, 5}
D) {x : x < 6}
Answer: A) {1, 2, 3, 4, 5}

57. Which of the following sets is infinite?

A) {x : x is a prime number}
B) {x : x is a digit in the decimal system}
C) {x : x is a natural number less than 100}
D) {x : x is a letter in the English alphabet}
Answer: A) {x : x is a prime number}

58. If A = {x : x is a natural number and x² < 16}, then A in roster form is:

A) {1, 2, 3, 4}
B) {1, 2, 3}
C) {1, 4, 9}
D) {2, 3, 4}
Answer: A) {1, 2, 3, 4}

59. The complement of the universal set U is:

A) U
B) φ
C) A′
D) {x : x ∉ U}
Answer: B) φ

60. If A = {x : x is a natural number and x ≤ 3}, then the cardinality of A is:

A) 1
B) 2
C) 3
D) 4
Answer: C) 3

61. The universal set for the set of all natural numbers is usually:

A) The set of integers
B) The set of rational numbers
C) The set of real numbers
D) The set of all natural numbers
Answer: D) The set of all natural numbers

62. If A = {1, 3, 5} and B = {2, 4, 6}, then A ∩ B is:

A) {1, 2, 3, 4, 5, 6}
B) φ
C) {3, 5}
D) {1, 3, 5, 2, 4, 6}
Answer: B) φ

63. The empty set is a subset of:

A) Only the universal set
B) Only finite sets
C) All sets
D) No set
Answer: C) All sets

64. Which of the following intervals represents all non-negative real numbers?

A) (-∞, ∞)
B) [0, ∞)
C) (-∞, 0)
D) (0, ∞)
Answer: B) [0, ∞)

65. If A = {a, e, i, o, u} and B = {x : x is a vowel in the English alphabet}, then A = B because:

A) A ∪ B = φ
B) A ∩ B = φ
C) Both sets have the same elements
D) A - B = φ
Answer: C) Both sets have the same elements

66. In set theory, a proper subset A of B means:

A) A = B
B) A ⊂ B but A ≠ B
C) A ⊄ B
D) A ∪ B = B
Answer: B) A ⊂ B but A ≠ B

67. If A = {x : x is an integer and x² ≤ 4}, then A in roster form is:

A) {0, 1, 2, 3, 4}
B) {1, 2, 3}
C) {0, -1, 1, -2, 2}
D) {0, 1, 2}
Answer: C) {0, -1, 1, -2, 2}

68. The set of all even integers is:

A) Infinite
B) Finite
C) A null set
D) A singleton set
Answer: A) Infinite

69. In a Venn diagram, the intersection of A and B is represented by:

A) The region inside the rectangle but outside the circles of A and B
B) The overlapping region of A and B
C) The region inside circle A but outside circle B
D) The region inside circle B but outside circle A
Answer: B) The overlapping region of A and B

70. The number of subsets of a set with n elements is:

A) n
B) 2ⁿ
C) n²
D) 2ⁿ - 1
Answer: B) 2ⁿ

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Friday, November 22, 2024

MCQ Mathematics Chapter 1: Sets, HS 1st year