Sets and Functions problems
- 1. If P = { 1, 2, 3 } and Q = { 4 , 5 } then what is the value of P × Q?
Options- A. { (4 , 5) , (1 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) }
- B. { (1 , 4) , (4 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) }
- C. { (1 , 4) , (1 , 5) , (2 , 4) , (3 , 5) , (3 , 4) , (3 , 5) }
- D. { (1 , 4) , (1 , 5) , (2 , 4) , (2 , 5) , (2 , 4) , (3 , 5) } Discuss
Correct Answer:
Explanation:
The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ).
Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) .
Given that P = { 1, 2, 3 } and Q = { 4 , 5 } then,
P x Q = { 1, 2, 3 } x { 4 , 5 }
? P x Q = { 1 } x { 4 , 5 } , { 2 } x { 4 , 5 } , { 3 } x { 4 , 5 }
? P x Q = { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 2 , 5 } , { 3 , 4 }, { 3 , 5 }
? P x Q = { (1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ), ( 3 , 5) }
- 2. If A and B have n elements in common, how many elements do A × B and B × A have in common?
Options- A. n
- B. n3
- C. n2
- D. None of these Discuss
Correct Answer: n2
Explanation:
NA
- 3. If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ? B) ? (A ? C).
Options- A. {c}
- B. {a}
- C. {b}
- D. {d} Discuss
Correct Answer: {c}
Explanation:
A ? B = {a, b, c} ? {c, d, e, f}
A ? B = { c }
A ? C = { a, b, c } ? { c, d, e }
A ? C = { c }
? (A ? B) ? (A ? C) = { c }.
- 4. If U = {a, b, c, d, e, f}, A = {a, b, c}, find (U ? A)?.
Options- A. U
- B. A
- C. ?
- D. None of these Discuss
Correct Answer: ?
Explanation:
U ? A = {a, b, c, d, e, f} ? {a, b, c} = {a, b, c, d, e, f} = U
(U ? A)? = ?.
- 5. Let A = {1, 3, 5} and B = {x: x is an odd natural number < 6}. Which of the following is false?
Options- A. A ? B
- B. B ? A
- C. A = B
- D. None of these Discuss
Correct Answer: None of these
Explanation:
NA
- 6. Let A = {?, {?}, 1, {1, ?}, 7}. Which of the following is false?
Options- A. ? ? A
- B. {?} ? A
- C. {1} ? A
- D. {7, ?,} ? A Discuss
Correct Answer: {1} ? A
Explanation:
NA
- 7. Write down the power set of the set {0}.
Options- A. ?
- B. {0}
- C. {?}
- D. {?, {0}} Discuss
Correct Answer: {?, {0}}
Explanation:
Let, A = {0}. The possible subsets of this set A are ? and {0}, so the power set of the given set A is P(A) = {f, {0}}.
- 8. For which of the following cases A and B are equivalent?
Options- A. A = {a, b, c,?, z}, B = {1, 2, 3,?, 24}
- B. A = {1/3,1/2,3/5}, B = { x : x = n/n+2, n ? N}
- C. A = {2, 4, 6}, B = {(2, 4), (4, 6), (2, 6)}
- D. A = { x : x = ( n3 - 1 )/ ( n3 + 1) , n ? W, n ? 3} , B = { 0,7/9,13/14} Discuss
Correct Answer: A = {2, 4, 6}, B = {(2, 4), (4, 6), (2, 6)}
Explanation:
NA
- 9. Find the cardinal number of the following set {x: x is a natural number ? 30 and is divisible by 7 or 11}
Options- A. 4
- B. 8
- C. 6
- D. 2 Discuss
Correct Answer: 6
Explanation:
NA
- 10. Which of the following sets is empty?
Options- A. A = {x: x ? n and x ? 1}
- B. B = {x: 3x + 1 = 0, x ? N}
- C. C = {x: x is an integer and -1 < x < 1}
- D. D = set of months of the year beginning with F. Discuss
Correct Answer: B = {x: 3x + 1 = 0, x ? N}
Explanation:
NA
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