Class XI SET THEORY MCQ SOLUTION | Sem 1

Chapter 1: Set Theory

SN Dey Class 11 Math Solution For Semester - 1

MCQ Questions

1. The number of subsets of a set containing four elements is—

(A) 8

(B) 8

(C) 16 (Answer)

(D) 64

Solution: If a finite set A has n elements, then its power set contains 2ⁿ elements. ∴ The number of subsets of a set with four elements is 2⁴ = 16.

2. The number of proper subsets of a set containing five elements is—

(A) 5

(B) 10

(C) 32

(D) 31 (Answer)

Solution: The number of proper subsets of a set with five elements is 2⁵ - 1 = 32 - 1 = 31.
[Note: A set is a subset of itself, but it is not a proper subset].

3. If x ∈ A ⇒ x ∈ B, then—

(A) A = B

(B) A ⊂ B

(C) A ⊆ B (Answer)

(D) B ⊆ A

4. If A ⊆ B and B ⊆ A, then—

(A) A = ϕ

(B) A ∩ B = ϕ

(C) A = B (Answer)

(D) None of these

5. If for two sets A and B, A ∪ B = A ∩ B, then—

(A) A ⊆ B

(B) B ⊆ A

(C) A = B (Answer)

(D) None of these

Solution: Let A = {2,4,6} and B = {2,4,6}.
∴ A ∪ B = A ∩ B = {2,4,6}.
Therefore, A ∪ B = A ∩ B is possible if and only if A = B.

6. A - B = ϕ if and only if—

(A) A ≠ B

(B) A ⊂ B (Answer)

(C) B ⊂ A

(D) A ∩ B = ϕ

Solution: Let A = { 2, 4, 6 } and B = { 1, 2, 3, 4, 5, 6, 7, 8 }.
The set (A - B) contains those elements that belong to A but do not belong to B. In this case, no such element is found.
Therefore, A - B = ϕ.

7. If A ∩ B = B, then—

(A) A ⊆ B

(B) B ⊆ A (Answer)

(C) A = B

(D) A = ϕ

Solution: Let A = { 10,20,30,40,50,60,70,80 } and B = { 30,40,50,60 }.
∴ B ⊆ A.
Now, A ∩ B = { 30,40,50,60 } = B.
∴ A ∩ B = B holds true only when B ⊆ A.

8. If A and B are two disjoint sets, then n(A ∪ B) =

(A) n(A) + n(B) (Answer)

(B) n(A) - n(B)

(C) 0

(D) None of these

Solution: Let A = { 2 , 4 , 6 , 8 , 10 } and B = { 1, 3, 5, 7, 9 }.
∴ A ∩ B = ϕ.
Thus, A and B are disjoint sets.
Now, (A ∪ B) = { 1,2,3,4,5,6,7,8,9,10 }.
∴ n(A ∪ B) = 10.
Also, n(A) + n(B) = 5 + 5 = 10.
∴ n(A ∪ B) = n(A) + n(B).

9. For any two sets A and B, n(A) + n(B) - n(A ∩ B) =

(A) n(A ∪ B) (Answer)

(B) n(A) - n(B)

(C) ϕ

(D) None of these

Solution: Let A = {a,b,c,d,e,f} and B = {e,f,g,h,i,j}.
∴ A ∪ B = {a,b,c,d,e,f,g,h,i,j}.
∴ n(A ∪ B) = 10.
Also, (A ∩ B) = {e,f}.
∴ n(A ∩ B) = 2.
∴ n(A) + n(B) - n(A ∩ B) = 6 + 6 - 2 = 12 - 2 = 10 = n(A ∪ B).

10. The dual identity of A ∪ U = U will be—

(A) A ∩ U = U

(B) A ∪ ϕ = ϕ

(C) A ∪ ϕ = A

(D) A ∩ ϕ = ϕ (Answer)

11. The dual identity of the identity A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) is—

(A) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Answer)

(B) A ∪ (B ∪ C) = (A ∪ B) ∪ (A ∪ C)

(C) A ∩ (B ∩ C) = (A ∩ B) ∩ (A ∩ C)

(D) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Solution: The operations of union and intersection of sets follow the Principle of Duality. According to this principle, if a set identity is true, the dual identity obtained by replacing ∪ with ∩ and ∩ with ∪ is also true.

12. Which of the following statements is true?

(A) A subset of an infinite set is always an infinite set.

(B) The set of even numbers greater than 889 is an infinite set. (Answer)

(C) The set of negative numbers greater than (-150) is an infinite set.

(D) A = {x : x is a real number and 0 < x ≤ 1} is a singleton set.

13. Which of the following statements is NOT true?

(A) If a ∈ A and a ∈ B, then A ⊆ B. (Answer)

(B) If A ⊆ B and B ⊆ C, then A ⊆ C.

(C) If A ⊆ B and B ⊆ A, then A = B.

(D) If A ∪ ϕ = ϕ, then A = ϕ.

14. Which of the following is the set of factors of the number 12?

(A) {2 , 3 , 4 , 6}

(B) {2,3,4,6,12}

(C) {2,3,4,6,8}

(D) {1,2,3,4,6,12} (Answer)

15. Which of the following sets is a null (empty) set?

(A) {0}

(B) {ϕ}

(C) {x : x is an integer and 1 < x < 2} (Answer)

(D) {x : x is a real number and 1 < x < 2}

Explanation: Since there is no integer between 1 and 2, the set {x : x is an integer and 1 < x < 2} is an empty set.

16. If B is the power set of set A, then which of the following is correct?

(A) A ⊃ B

(B) B ⊃ A

(C) A ∈ B (Answer)

(D) A = B

Explanation: The power set of A is the set containing all subsets of A. Since every set is a subset of itself, set A is an element of its power set B. ∴ A ∈ B.

17. If x ∈ A ∪ B, then which of the following is correct?

(A) x ∈ A

(B) x ∈ B

(C) x ∈ A or x ∈ B (Answer)

(D) x ∈ A and x ∈ B

18. If x ∈ A ∩ B, then which of the following is correct?

(A) x ∈ A and x ∈ B (Answer)

(B) x ∈ B

(C) x ∈ A or x ∈ B

(D) x ∉ A

Explanation: x ∈ A ∩ B means x is present in A as well as in B. Therefore, x ∈ A ∩ B ⇒ x ∈ A and x ∈ B.

19. If A = {2,4,6,8}, then which of the following is correct?

(A) {2,4} ∈ A

(B) {2,4} ⊆ A

(C) {2,4} ⊂ A (Answer)

(D) {2,4} ∈ A^c

Explanation: Since the set {2,4} is a proper subset of set A, therefore {2,4} ⊂ A.

20. Which of the following statements is correct?

(A) {a} ∈ {a,b,c}

(B) a ∉ {a,b,c}

(C) a ⊂ {a,b,c}

(D) {a} ⊂ {a,b,c} (Answer)

Explanation: Since the set {a} is a proper subset of the set {a,b,c}, therefore {a} ⊂ {a,b,c}.

21. Which two of the four defined sets below are equal?

(i) A = {0}
(ii) B = {Φ}
(iii) C = {x : x is a perfect square and 2 ≤ x ≤ 6}
(iv) D = {x : x is an integer and -1 < x < 1}

(A) (i) and (iv) (Answer)

(B) (ii) and (iv)

(C) (ii) and (iii)

(D) (iii) and (iv)

Explanation: D = {x : x is an integer and -1 < x < 1} = {0} = A, because 0 is the only integer lying between -1 and 1.

22. Which of the following defined sets is an empty set?

(A) A = {x : x is a perfect cube of an integer and 2 ≤ x ≤ 7} (Answer)

(B) B = { 0 }

(C) C = { Φ }

(D) D = {x : x is an integer and 2 < x ≤ 3}

Explanation: There is no perfect cube integer between 2 and 7. Thus, A = Φ (null set).

23. Which of the following defined sets is an infinite set?

(A) A = {x : x is an integer and -1 ≤ x < 1}

(B) B = Set of even negative numbers greater than (-100)

(C) C = Set of positive integers less than 100

(D) D = {x : x is real and -1 ≤ x < 1} (Answer)

Explanation: Since there are infinitely many real numbers between -1 and 1, set D is an infinite set.

24. The power set of the set A = { {1}, {2,3} } is—

(A) { Φ , {1}, {2,3}, A}

(B) { {1}, {2,3}, A}

(C) { Φ , {{1}} , {{2,3}} , A} (Answer)

(D) { Φ, {{1}} , {2} , {3} , {{2,3}}, A}

Explanation: The power set contains all the subsets of A. Here, Φ is a subset of any set, and A is a subset of itself. Taking elements {1} and {2,3} as individual elements, their subsets become {{1}} and {{2,3}}. Thus, C is the correct answer.

25. If A = {1,2,3,4} , B = { 2,4,5,8 } , C = { 3,4,5,6,7 }, then—

(A) A ∪ B = {1,2,3,4,5}

(B) B ∩ C = {4,5} (Answer)

(C) A ∪ (B ∪ C) = {5,6,7}

(D) A ∪ (B ∩ C) = {1,2,3,4,5,6,7,8}

Explanation: The common elements between set B and set C are 4 and 5. ∴ B ∩ C = {4,5}.

26. If P = {a,b,c,d,e} and Q = {a,e,i,o,u}, then—

(A) P ⊂ Q

(B) Q ⊂ P

(C) P ∩ Q = {a,e} (Answer)

(D) P ∪ Q = {a,b,l,c,d,u}

Explanation: The common elements between set P and set Q are a and e. ∴ P ∩ Q = {a,e}.

27. Let universal set A = {a,b,c} , B = {a,b} , C = {a,b,d} , D = {c,d} and E = {d}; which of the following is true?

(A) {a} ∈ A

(B) D ⊄ E

(C) D ⊂ B

(D) {a} ⊂ A (Answer)

28. Let the universal set be S = {1,2,3,4,5}, and let A = {3,4,5} and B = {1,4,5} be its two subsets. Then (A ∪ B)' =

(A) {2} (Answer)

(B) {1,2}

(C) {1,3,4,5}

(D) None of these

Solution: A ∪ B = { 3, 4, 5, 1 }
∴ (A ∪ B)' = S - (A ∪ B) = {1,2,3,4,5} - {3,4,5,1} = {2}.

29. If A = {1,2,3,4} , B = {2,3,4,5} , C = {1,3,4,5,6,7}, then what is the value of A - (B ∩ C)?

(A) {1}

(B) {2}

(C) {1,2} (Answer)

(D) {{1},{2}}

Solution: (B ∩ C) = {x : x ∈ B and x ∈ C} = {Elements present in both B and C} = {3,4,5}.
A - (B ∩ C) = {Elements present in A but not in (B ∩ C)} = {1,2}.

30. If the universal set is S = {1,2,4,8,16,32}, and A = {1,2,8,32}, B = {4,8,32} are its two subsets, then (A ∪ B)^c =

(A) {1,2,4,8,32}

(B) {16} (Answer)

(C) {8,32}

(D) {1,2,4,16}

Solution: (A ∪ B) = {All those numbers which belong to A or belong to B or belong to both} = {1,2,4,8,32}
∴ (A ∪ B)^c = {All those numbers which belong to S but do not belong to (A ∪ B)} = {16}

31. If P = {a,b,c,d,e,f} and Q = {a,c,e,f}, then (P-Q) ∪ (P ∩ Q) will be –

(A) Q

(B) P ∪ Q

(C) P ∩ Q

(D) P (Answer)

Solution: (P-Q) = {All those letters which belong to P but do not belong to Q} = {b,d}
(P ∩ Q) = {All those letters which belong to both P and Q} = {a,c,e,f}
∴ (P-Q) ∪ (P ∩ Q) = {a,b,c,d,e,f} = P.

32. If P = {θ : sin θ - cos θ = √2 cos θ} and Q = {θ : sin θ + cos θ = √2 sin θ}, then—

(A) P ⊂ Q

(B) Q ⊂ P

(C) P = Q (Answer)

(D) P ∩ Q = Φ

Solution:
sin θ - cos θ = √2 cos θ
or, sin θ = cos θ + √2 cos θ
or, sin θ = cos θ (1 + √2)
or, (√2 - 1) sin θ = (√2 + 1)(√2 - 1) cos θ
or, (√2 - 1) sin θ = {(√2)² - 1} cos θ
or, (√2 - 1) sin θ = (2 - 1) cos θ
or, √2 sin θ - sin θ = cos θ
or, sin θ + cos θ = √2 sin θ
∴ P = Q

33. Given A = {1,2,3,4,5} and (B ∪ C) = {3,4,6}, then (A-B) ∩ (A-C) =

(A) {1,2}

(B) {2,5}

(C) {1,2,5} (Answer)

(D) {1,2,6}

Solution: (A-B) ∩ (A-C) = A - (B ∪ C) = {All those numbers which belong to A but do not belong to (B ∪ C)} = {1,2,5}

34. Let the universal set be S = {a,b,c,d,e} and let A = {a,b,d} and B = {b,d,e} be its two subsets. Then (A ∪ B)^c =

(A) {a,c,e}

(B) {c} (Answer)

(C) {a,b,d,e}

(D) None of these

Solution: (A ∪ B) = {All those letters which belong to A or belong to B} = {a,b,d,e}
∴ (A ∪ B)^c = {All those letters which belong to S but do not belong to (A ∪ B)} = {c}

35. Let the universal set be S = {1,2,3,4,5,6} and A ∪ B = {2,3,4}; then A^c ∩ B^c =

(A) {1,5,6} (Answer)

(B) {2,3,4}

(C) Φ

(D) None of these

Solution:
A^c ∩ B^c = (A ∪ B)^c [De Morgan's Laws]
= {All those numbers which belong to S but do not belong to (A ∪ B)} = {1, 5, 6}

36. If the set of natural numbers is ℕ and aℕ = {ax : x ∈ ℕ}, then (3ℕ ∩ 7ℕ) =

(A) ℕ

(B) 5ℕ

(C) 21ℕ (Answer)

(D) None of these

Solution: aℕ = {ax : x ∈ ℕ}
∴ 3ℕ = {3,6,9,12,18, 21, 24,27,30,33,36,39, 42, ...}
and 7ℕ = {7,14, 21, 28,35, 42, 49,56, ...}
∴ (3ℕ ∩ 7ℕ) = {All those numbers which belong to both 3ℕ and 7ℕ} = 21ℕ

37. Let the set of all integers be ℤ, and A = {x : x = 6n, n ∈ ℤ} and B = {x : x = 4n, n ∈ ℤ}; then A ∩ B =

(A) {x: x=2n, n∈ℤ}

(B) {x: x=12n, n ∈ ℤ} (Answer)

(C) {x : x=24n, n ∈ ℤ}

(D) None of these

Solution: A = {..., -24, -18, -12, -6, 0, 6, 12, 18, 24, ...}
B = {..., -24, -20, -16, -12, -8, -4, 0, 4, 8, 12, 16, 20, 24, ...}
∴ A ∩ B = {..., -24, -12, 0, 12, 24, ...} = {x : x = 12n, n ∈ ℤ}

38. For any two sets A and B, which of the following statements is NOT true?

(A) (B-A) ∩ A = Φ

(B) A^c - B^c = B - A

(C) A-B = A-(A ∩ B)

(D) A-B = A^c ∩ B (Answer)

Solution: Only A-B = A^c ∩ B is not true because A-B = A ∩ B^c ≠ A^c ∩ B.
• (B-A) ∩ A = (B ∩ A^c) ∩ A = B ∩ (A^c ∩ A) = B ∩ Φ = Φ (True)
• A^c - B^c = A^c ∩ (B^c)^c = A^c ∩ B = B - A (True)
• A-(A ∩ B) = A ∩ (A ∩ B)^c = A ∩ (A^c ∪ B^c) = (A ∩ A^c) ∪ (A ∩ B^c) = Φ ∪ (A ∩ B^c) = A - B (True)

39. Let A, B, C be three given sets. Which of the following statements is true?

(A) If B ∈ A and x ∈ B, then x ∈ A

(B) If B ⊂ A and A ∈ C, then B ∈ C

(C) If A ∉ B and B ∉ C, then A ∉ C

(D) None of these. (Answer)

40. In a class, there are 70 students, each of whom studies either English or Hindi or both subjects. 45 students study English and 30 study Hindi. How many students study both subjects?

(A) 5 students (Answer)

(B) 15 students

(C) 25 students

(D) 40 students

Solution: Let the number of students studying English be E and Hindi be H. ∴ n(E ∪ H) = 70, n(E) = 45, n(H) = 30.
n(E ∪ H) = n(E) + n(H) - n(E ∩ H)
or, 70 = 45 + 30 - n(E ∩ H)
or, 70 = 75 - n(E ∩ H)
or, n(E ∩ H) = 75 - 70 = 5.

41. In a survey of 1003 families in Kolkata, it was found that 63 families had neither radio nor TV; 794 families had radios and 187 families had TVs. How many families had both radio and TV?

(A) 85

(B) 22

(C) 43

(D) 41 (Answer)

Solution: n(R) = 794 , n(T) = 187 and n(R ∪ T)^c = 63.
∴ n(R ∪ T) = 1003 - 63 = 940.
n(R ∪ T) = n(R) + n(T) - n(R ∩ T)
or, 940 = 794 + 187 - n(R ∩ T)
or, n(R ∩ T) = 981 - 940 = 41.

42. A market research team surveyed 1000 consumers and reported that 720 consumers liked product A and 450 consumers liked product B. What is the least number that must have liked both products?

(A) 585

(B) 170 (Answer)

(C) 270

(D) 280

Solution: n(A) = 720 , n(B) = 450 , n(A ∪ B) = 1000.
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
or, 1000 = 720 + 450 - n(A ∩ B)
or, n(A ∩ B) = 1170 - 1000 = 170.

43. The number of elements in two sets A and B are p and q respectively. If the number of subsets of set A is 56 more than the number of subsets of set B, then the values of p and q are respectively -

(A) 3 , 6

(B) 6 , 3 (Answer)

(C) 5 , 7

(D) 3 , 4

Solution: Number of subsets of A = 2^p, number of subsets of B = 2^q.
According to the condition: 2^p = 2^q + 56 ⇒ 2^p - 2^q = 56.
If p = 6 and q = 3: 2⁶ - 2³ = 64 - 8 = 56. Thus, p = 6 and q = 3.

44. If the number of elements in two finite sets A and B are m and n respectively, then the maximum number of elements in A ∪ B is –

(A) mn

(B) m-n

(C) m+n (Answer)

(D) None of these

Solution: If A and B are two disjoint sets, then n(A ∪ B) = n(A) + n(B) = m+n [Maximum case]. If they are not disjoint, n(A ∪ B) = m+n - n(A ∩ B).

45. Let the universal set be S = {x : 0 < x ≤ 10} and its two subsets be A = {x : 2 ≤ x < 5} and B = {x : 3 < x < 7}; then A ∩ B =

(A) {x : 3 ≤ x ≤ 5}

(B) {x : 3 < x < 5} (Answer)

(C) {x : 2 ≤ x < 7}

(D) {x : 2 < x < 3}

Solution: S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 3, 4} and B = {4, 5, 6}
∴ A ∩ B = {4} = {x : 3 < x < 5}.

46. If P = {p,q,r,s,t,u} and Q ∩ R = {q, r, v, w}, then (P-Q) ∪ (P-R) =

(A) {p,s,t}

(B) {p,s,t,u,w}

(C) {s,t,u}

(D) {p,s,t,u} (Answer)

Solution:
(P-Q) ∪ (P-R) = (P ∩ Q^c) ∪ (P ∩ R^c)
= P ∩ (Q^c ∪ R^c) [Distributive Law]
= P ∩ (Q ∩ R)^c [De Morgan's Laws]
= {P - (Q ∩ R)} = {p,s,t,u}

47. If A, B, C are three subsets of the universal set S, where S = {1,2,3,4,5,6,7} , A = {1,3,5,6} , B ∩ C = {1,2,6}, then (B^c ∪ C^c) =

(A) {3 ,4,5}

(B) {1,3,4,5,7}

(C) {3,4,5,7} (Answer)

(D) None of these

Solution: By De Morgan's Law: (B^c ∪ C^c) = (B ∩ C)^c = S - {1,2,6} = {3, 4, 5, 7}.

48. If U = {a,b,c,d,e,f} is the universal set and A, B, C are three subsets of U, where A = {a,c,d} and B ∪ C = {a,d,c,f}, then (A ∩ B) ∪ (A ∩ C) =

(A) {b,e}

(B) {a,b,c,d}

(C) {a,c,d} (Answer)

(D) {c,d }

Solution: (A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C)
= {a,c,d} ∩ {a,d,c,f} = {a,c,d}.

49. For any three sets A, B, C, which of the following is true?

(A) A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)

(B) A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)

(C) A - (B ∩ C) = (A ∩ B) - C

(D) A - (B ∪ C) = (A - B) ∩ (A - C) (Answer)

Solution: A - (B ∪ C) = A ∩ (B ∪ C)^c
= A ∩ (B^c ∩ R^c) = (A ∩ B^c) ∩ (A ∩ C^c) = (A - B) ∩ (A - C).

50. A ∩ (B-A) =

(A) A ∩ B

(B) A ∪ B

(C) Φ (Answer)

(D) None of these

Solution: A ∩ (B-A) = A ∩ (B ∩ A^c)
= A ∩ (A^c ∩ B) = (A ∩ A^c) ∩ B = Φ ∩ B = Φ.

51. If two sets A and B are defined as A = {(x,y) : y = 1/x, x ≠ 0, x ∈ ℝ} and B = {(x,y) : y = -x, x ∈ ℝ}, then:

(A) A ∩ B = A

(B) A ∩ B = B

(C) A ∩ B = Φ (Answer)

(D) A ∪ B = A

Solution: A represents a hyperbola in the 1st and 3rd quadrants. B represents a line running through the 2nd and 4th quadrants. They never intersect. Therefore, A ∩ B = Φ.

52. In a class of 60 students, 25 play cricket, 20 students play tennis and 10 students play both games. Then the number of students who play neither game will be –

(A) 0

(B) 25 (Answer)

(C) 35

(D) 45

Solution: n(C) = 25, n(T) = 20, n(C ∩ T) = 10.
Only Cricket = 25 - 10 = 15.
Only Tennis = 20 - 10 = 10.
Neither Game = 60 - (15 + 10 + 10) = 60 - 35 = 25.