Sets and Functions problems
- 1. Which of the following sets is finite?
Options- A. {x: x ? N and x is a prime number}
- B. {x: x is a quadrilateral on a plane}
- C. {x: x ? N and x2 ? 25 ? 0}
- D. {x: x ? N and x is a multiple of 3} Discuss
Correct Answer: {x: x ? N and x2 ? 25 ? 0}
Explanation:
NA
- 2. Which of the following sets is infinite?
Options- A. {x: x is a prime number, x is even}
- B. Set of all river in India
- C. Set of all concentric circles
- D. None of above Discuss
Correct Answer: Set of all concentric circles
Explanation:
NA
- 3. Find the cardinal number of the following set :
{x : x is a letter of the word 'ASSASSINATION'}
Options- A. 4
- B. 6
- C. 8
- D. 2 Discuss
Correct Answer: 6
Explanation:
NA
- 4. Which of the following statements is false for the sets A, B and C, where:
A = {x | x is letter of the word 'BOWL'}
B = {x | x is a letter of the word 'ELBOW'}
C = {x | x is a letter of the word 'BELLOW'}
Options- A. A ? B
- B. B ? C
- C. B = C
- D. B is a proper subset of C. Discuss
Correct Answer: B is a proper subset of C.
Explanation:
NA
- 5. Which of the following pairs of sets are disjoint?
(i) {1, 2, 3, 4 } and { x : x is a natural number and 4 ? x ? 6 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) { x : x is an even integer } and { x : x is an odd integer }.
Options- A. (i)
- B. (ii)
- C. (iii)
- D. None of these Discuss
Correct Answer: (iii)
Explanation:
(a) {x: x is a natural number and 4 ? x ? 6} = {4, 5, 6}. Now, {1, 2, 3, 4} and {4, 5, 6} have one element 4 common. Therefore, the given two sets are not disjoint.
(b) The sets {a, e, i, o, u} and {c, d, e, f} have one element e as common. Then, the given two sets are not disjoint.
(c) The sets {x: x is an even integer} and {x: x is an odd integer} have no element as common and therefore they are disjoint sets.
- 6. If X and Y are two sets such that X ? Y has 18 elements, X has 8 elements, and Y has 15 elements, how many elements does X ? Y have?
Options- A. 5
- B. 7
- C. 9
- D. 11 Discuss
Correct Answer: 5
Explanation:
Given in the question ,
n(X ?Y) = 18, n(X) = 8, n(Y) = 15.
Using the formula
n(X ? Y) = n(X) + n(Y) ? n(X ? Y), we get n(X ? Y) = 8 + 15 ? 18 = 5.
- 7. If A and B are two sets such that A has 40 elements, A ? B has 60 elements and A ? B has 10 elements, how many elements does B have?
Options- A. 40
- B. 30
- C. 45
- D. 50 Discuss
Correct Answer: 30
Explanation:
Given in the question ,
n(A) = 40, n(A ? B) = 60 and n(A ? B) = 10.
As we know the formula,
n(A ? B) = n(A) + n(B) ? n(A ? B)
Putting these values in the formula, we get
60 = 40 + n(B) ? 10
? n(B) = 30.
- 8. In a group of 1000 people, there are 750 people who can speak Hindi and 400 who can speak English. How many can speak Hindi only?
Options- A. 600
- B. 650
- C. 750
- D. 800 Discuss
Correct Answer: 600
Explanation:
Given in the question,
n(H ? E) = 1000, n(H) = 750, n(E) = 400
Use the below formula,
n(H ? E) = n(H) + n(E) ? n(H ? E)
We get 1000 = 750 + 400 ? n(H ? E)
? n(H ? E) = 1150 ? 1000 = 150.
Number of People who can speak Hindi only
= n(H ? E?) = n(H) ? n(H ? E)
= 750 ? 150 = 600.
- 9. In a town with a population of 5000, 3200 people are egg-eaters, 2500 meat eaters and 1500 eat both egg and meat. How many are pure vegetarians?
Options- A. 600
- B. 800
- C. 900
- D. 850 Discuss
Correct Answer: 800
Explanation:
Let, E be the set of people who are egg-eaters and M be the set of people who are meat-eaters.
As per given question,
n(E) = 3200, n(M) = 2500, n(E ? M) = 1500.
Use the formula
n(E ? M) = n(E) + n(M) ? n(E ? M)
= 3200 + 2500 ? 1500
= 5700 ? 1500 = 4200.
? Number of pure vegetarians = n(U) ? n(E ? M)
Number of pure vegetarians = 5000 ? 4200 = 800.
- 10. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?
Options- A. 19
- B. 17
- C. 23
- D. 21 Discuss
Correct Answer: 19
Explanation:
Let P = set of people who like coffee and Q = set of people like tea.
Then, P ? Q = set of people who like at least one of the two drinks.
And P ? Q = set of people who like both the drinks.
Given in the question,
n(P) = 37, n(Q) = 52, n(P ? Q) = 70.
Use the below formula,
n(P ? Q) = n(P) + n(Q) ? n(P ? Q)
Put the value from the given question,
70 = 37 + 52 ? n(P ? Q)
? n(P ? Q) = 89 ? 70 = 19.
? 19 people like both coffee and tea.
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