Sets and Functions problems
- 1. Which of the following sets is finite?
Options- A. The set of the months of a year
- B. {1, 2, 3,?}
- C. {1, 2, 3,? 90, 100}
- D. The set of lines which are parallel to x-axis Discuss
Correct Answer: The set of the months of a year
Explanation:
NA
- 2. Which of the following sets is finite?
Options- A. I = { x : x ? Z and x2 ? 2x ? 3 = 0 }
- B. B = The set of natural numbers which are divisible by 2
- C. C = The set of lines passing through a point.
- D. D = { x : x ? Z and x > - 5 } Discuss
Correct Answer: I = { x : x ? Z and x2 ? 2x ? 3 = 0 }
Explanation:
(a) A = { x : x ? Z and x2 ? 2x ? 3 = 0} = {3, -1}.
? A is a finite set.
(b) B = The set of natural numbers divisible by 2 = { 2, 4, 6, 8, 10,?}.
? B is an infinite set.
(c) Since infinite number of lines pass through a point, C is an infinite set.
(d) D = { -4, -3, -2,?}. Clearly D is an infinite set.
- 3. Which of the following sets is non-empty?
Options- A. A = set of odd natural numbers divisible by 2
- B. B = {x : x + 5 = 0, x ? N}
- C. C = set of even prime numbers
- D. D = {x : 1 < x < 2, x ? N}, Discuss
Correct Answer: C = set of even prime numbers
Explanation:
A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3, ...}
As per given option.
(a) As no odd natural number is divisible by 2, the set A is empty.
(b) Since no natural number satisfies the equation x + 5 = 0,
? B = ?
(c) Since 2 is an even prime number, i.e., C = {2}, C is not an empty set.
(d) Since there is no natural number between 1 and 2, D is an empty set.
- 4. Write down the power set of C = {1, {2}}.
Options- A. {?, {1}, {{2}}}
- B. {?, {1}, {{2}}, {1, {2}}}
- C. {{1}, {{2}}, {1, {2}}
- D. None of these Discuss
Correct Answer: {?, {1}, {{2}}, {1, {2}}}
Explanation:
NA
- 5. Which of the following statements is true?
Options- A. Every subset of a finite set is finite.
- B. Every subset of an infinite set is infinite.
- C. Every subset of an infinite set is finite.
- D. A proper subset of a finite set is equivalent to the set itself. Discuss
Correct Answer: Every subset of a finite set is finite.
Explanation:
NA
- 6. 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. C
- D. None of these Discuss
Correct Answer: C
Explanation:
B = {5, 10, 15,?}
C = {10, 20, 30,?}
B ? C = {5, 10, 15,?} ? {10, 20, 30,?}
B ? C = {5, 10, 15,?}
? A ? (B ? C) = {2, 4, 6,?} ? {5, 10, 15,?} = {10, 20, 20,?} = C.
- 7. If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then compute (A ? B)?.
Options- A. {2, 3, 5, 7, 9, 11, 12}
- B. {2, 4, 6, 8, 10, 11, 12}
- C. {2, 4, 6, 8, 9, 10, 11}
- D. None of these Discuss
Correct Answer: None of these
Explanation:
A ? B is a set of member which belong to A but do not belong to B
? A ? B = {3, 5, 7, 9, 11} - {7, 8, 9, 10, 11}
A ? B = {3, 5}
According to formula,
(A ? B)? = U - (A ? B)
? (A ? B)? = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} - {3, 5}
(A ? B)? = {2, 4, 6, 7, 8, 9, 10, 11}.
- 8. If S and T are two sets such that S has 21 elements, T has 32 elements, and S ? T has 11 elements, how many elements does S ? T have?
Options- A. 52
- B. 32
- C. 42
- D. None of these Discuss
Correct Answer: 42
Explanation:
Given in the question,
n(S) = 21, n(T) = 32, n(S ? T) = 11, n(S ? T) = ?
Using the formula,
n(S ? T) = n(S) + n(T) ? n(S ? T) = 21 + 32 ? 11 = 42
Hence, S ? T has 42 elements.
- 9. Let A = {1, 2}, B = {2, 3}. Evaluate A × B.
Options- A. {(2, 1), (3, 1), (2, 3)}
- B. {(1, 2), (1, 3), (2, 3)}
- C. {(1, 2), (1, 3), (2, 2), (2, 3)}
- D. None of these Discuss
Correct Answer: {(1, 2), (1, 3), (2, 2), (2, 3)}
Explanation:
A × B = {1, 2} × {2, 3}
= {(1, 2), (1, 3), (2, 2), (2, 3)}.
- 10. The Cartesian product A × A has 9 elements and two of them are (-1, 0) and (0, 1). Find the set A?
Options- A. A = {0, 1}
- B. A = {-1, 0, 1}
- C. A = {0, -1}
- D. A = {1, -1,} Discuss
Correct Answer: A = {-1, 0, 1}
Explanation:
NA
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