Sets and Functions problems
- 1. Let A = {1, 2, {3, 4}, 5}. Which of the following statements is true?
Options- A. {3, 4} ? A
- B. {3, 4} ? A
- C. 1 ? A
- D. {1, 2, 5} ? A Discuss
Correct Answer: {3, 4} ? A
Explanation:
NA
- 2. Find the power set of A = {{a, b}, c}.
Options- A. {?, {a, b}, {c}}
- B. {A, {a, b}, {c}}
- C. {?, A, {a, b}, {c}}
- D. None of these Discuss
Correct Answer: {?, A, {a, b}, {c}}
Explanation:
Let, A = {{a, b}, c}. To determine P(A) : Since A contains two elements {a, b}, c, P(A) will contain 22 = 4 elements.
The elements of P(A) are ?, A, {a, b}, {c}.
- 3. In which of the following cases, A = B?
Options- A. A = {12, 14, 16}, B = {16, 18, 20}
- B. A = ?, b = {}
- C. A = {x: x ? W and x < 1}, B = ?
- D. A = {x: x is a day of the week beginning with S}, B = {Sunday} Discuss
Correct Answer: A = ?, b = {}
Explanation:
NA
- 4. Find the cardinal number of the following set {x: x = 2n, n ? N, 4 ? x ? 11}
Options- A. 8
- B. 6
- C. 12
- D. 4 Discuss
Correct Answer: 4
Explanation:
NA
- 5. Which of the following pairs of sets are not equal?
Options- A. (a) A = {1, 3, 3, 1}, B = {1, 4}
- B. (b) A = {x: x + 2 = 2}, B = {0}
- C. (c) A = {1, 3, 4, 4}, B = {3, 1, 4}
- D. (d) A = {1,1/2,1/3,... }, B { 1/n : n? N } . Discuss
Correct Answer: (a) A = {1, 3, 3, 1}, B = {1, 4}
Explanation:
(a) A = {1, 3}; B = {1, 4}. A and B have different elements,
? A ? B.
(b) A = {x: x + 2 = 2} = {0}; B = {0}. A and B have same elements,
? A = B.
(c) A = {1, 3, 4, 4} = {1, 3, 4}; B = {3, 1, 4}. A and B have same elements,
? A = B.
(d) A = { 1,1/2,1/3, ................ } ;
B = {1/n : n ? N} = { 1/1,1/2,1/3,...} .
A and B have same elements, ? A = B.
- 6. Which of the following pairs of sets are not equivalent?
Options- A. A = {2, 4, 6, 8}, B = {u, v, w, x}
- B. A = {a, b, c}, B = {?, ?, ?, ?, ?}
- C. A = {}, B = ?
- D. A = {x: x = 2n, n ? N}, B = {x: x = 2n + 1, n ? N}. Discuss
Correct Answer: A = {a, b, c}, B = {?, ?, ?, ?, ?}
Explanation:
NA
- 7. Let A = {x: x ? N ? x is a multiple of 2};
B = {x: x ? N ? x is a multiple of 5};
C = {x: x ? N ? x is a multiple of 10};
Describe the set (A ? B) ? C,
Options- A. A
- B. B
- C. A ? B
- D. C Discuss
Correct Answer: C
Explanation:
A = {2, 4, 6,?}
B = {5, 10, 15,?}
C = {10, 20, 30,?}
? (A ? B) = {2, 4, 6?} ? {5, 10, 15,?} = {10, 20, 30,?} = C
? (A ? B) ? C = C ? C = C.
- 8. If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11},
A = {2, 4, 7},
B = {3, 5, 7, 9, 11} and
C = {7, 8, 9, 10, 11}.
compute: (A ? U) ? (B ? C)
Options- A. {7}
- B. {9}
- C. {6}
- D. {5} Discuss
Correct Answer: {7}
Explanation:
A ? U = {2, 4, 7} ? {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
A ? U = {2, 4, 7};
B ? C = {3, 5, 7, 9, 11} ? {7, 8, 9, 10, 11}.
B ? C = {3, 5, 7, 8, 9, 10, 11}.
Then (A ? U) ? (B ? C) = {2, 4, 7,} ? {3, 5, 7, 8, 9, 10, 11}
(A ? U) ? (B ? C) = {7}.
- 9. In a class of 100 students, the number of students passed in English only is 46, in Maths only is 46, in Commerce only is 58. The number who passed in English and Maths is 16, Maths and Commerce is 24 and English and Commerce is 26, and the number who passed in all the subjects is 7. Find the number of the students who failed in all the subjects.
Options- A. 9
- B. 8
- C. 10
- D. None of these Discuss
Correct Answer: 9
Explanation:
No. of students who passed in one or more subjects = 11 + 9 + 13 + 17 + 15 + 19 + 7 = 91.
No. of students who failed in all the subjects = 100 ? 91 = 9.
- 10. If A = {a, b}, B = {2, 3, 5, 6, 7} and C = {5, 6, 7, 8, 9}, find A × (B ? C).
Options- A. A
- B. ?
- C. {(5, a), (6, a), (7, a), (5, b), (6, b), (7, b)}
- D. {(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)} Discuss
Correct Answer: {(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)}
Explanation:
We have,
(B ? C) = {2, 3, 5, 6, 7} ? {5, 6, 7, 8, 9}
= {5, 6, 7}
? A × (B ? C) = {a, b} × {5, 6, 7}
= {(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)}.
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