Question 1

If A = {a, d}, B = {b, c, e} and C = {b, c, f}, then A × (B ∩ C) =

Accepted Answer

Given in the question, we can can calculate, (B ∩ C) = {b, c, e} ∩ {b, c, f} = {b, c} ∴ A × (B ∩ C) = {a, d} × {b, c} = {(a, b), (a, c), (d, b), (d, c)} Also, (A × B) ∩ (A × C) = {(a, b), (a, c), (a, b), (d, c)} ∴ A × (B ∩ C) = (A × B) ∩ (A × C).

Question 2

If A = {a, d}, B = {b, c, e} and C = {b, c, f}, then A × (B - C) =

Accepted Answer

we can calculate below as per given question, we can calculate below as per given question, (B − C) = { b, c, e } – { b, c, f } = { e } A × (B − C) = { a, d } x { e } (A × B) = { a, d } x { b, c, e } (A × C) = { a, d } x { b, c, f } ∴ A × (B − C) = {(a, e), (d, e)} ...........…(1) Also, (A × B) – (A × C) = {(a, e), (d, e)} ..........…(2) Hence, from (1) and (2), we have A × (B − C) = (A × B) – (A × C).

Question 3

If A = {1, 2, 3}, B = {2, 3, 4}, C = {1, 3, 4} and D = {2, 4, 5}, then (A × B) ∩ (C × D) =?

Accepted Answer

(A × B) = {1, 2, 3} × {2, 3, 4} = {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} (C × D) = {1, 3, 4} × {2, 4, 5} = {(1, 2), (1, 4), (1, 5), (3, 2), (3, 4), (3, 5), (4, 2), (4, 4), (4, 5)} ∴ (A ∩ B) ∩ (C × D) = {(1, 2), (1, 4), (3, 2), (3, 4)} Also, (A ∩ C) = {1, 3} and (B ∩ D) = {2, 4}. Therefore, (A ∩ C) × (B ∩ D) = {(1, 2), (1, 4), (3, 2), (3, 4)} Hence, (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).

Question 4

In a group of 50 people, 35 speak Hindi, 25 speak both Hindi and English and all the people speak Hindi or English or both. The number of people who speak English only is:

Accepted Answer

Draw a figure and calculate. Number of people speak English only = 50 – (10 + 25) = 15

Question 5

There are 80 families in a small extension area. 20 per cent of these families own a car each. 50 per cent of the remaining families own a motor cycle each. How many families in that extension do not own any vehicle?

Accepted Answer

Given in the question, 20% of 80 = 20 * 80 / 100 = 16 Remaining 50% = (80 – 16) × 50 / 100 = 32 No. of families not owning any vehicle = 80 – (32 + 16) = 80 – 48 = 32

Question 6

In an examination, 30% of the total students failed in Hindi, 45% failed in English and 20% failed in both the subjects. Find the percentage of those who passed in both the subjects.

Accepted Answer

Let the number of students be 100. Number of students who failed in Hindi is 30%. n(H) = 30 Number of students who failed in English is 45%. ∴ n(E) = 45 Number of students who failed in both the subjects is 20%. n(H ∪ E) = 20 Applying the rule, n(H ∪ E) = n(H) + n(E) - n(H ∩ E) = 30 + 45 - 20 = 55 Percentage of students who failed in Hindi or English or both the subjects = 55% Number of students who passed in both the subjects = 100 - 55 = 45%

Question 7

If A = {1, 2, 3, 4, 5} and B = {2, 4, 6} then what is the value of A - B?

Accepted Answer

A new set having those elements which are in A but not in B is said to be the difference of sets A and B and it is denoted by A ? B. Given that in question, A = {1, 2, 3, 4, 5} and B = {2, 4, 6} So the numbers which are in A and but not in B = {1, 3, 5} ? A ? B = {1, 3, 5}

Question 8

Write down the power set of A = {8, 9}.

Accepted Answer

NA

Question 9

Which of the following pairs of sets are comparable?

Accepted Answer

(a) 1∈ A but 1∉ B and 6 ∈ B but 6 ∉ A. ∴ A and B are not comparable. (b) A = {x: x ∈ N and x ≤ 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Clearly, A ⊂ B ⇒ A and B are comparable. (c) {4, 5} ∈ A but {4, 5} ∉ B and 4 ∈ B but 4 ∉ A. ∴ A and B are not comparable.

Question 10

In a class, 50 students play cricket, 20 students play football and 10 play both cricket and football. How many play at least one of these two games?

Accepted Answer

Given: n(C) = 50, n(F) = 20, n(C ∪ F) = 10. Number of students playing at least one of these two games = n (C ∪ F) = n (C) + n (F) – n (C ∩ F) = 50 + 20 – 10 = 60.

Question 11

Which of the following sets is finite?

Accepted Answer

NA

Question 12

Which of the following sets is infinite?

Accepted Answer

NA

Question 13

Find the cardinal number of the following set : {x : x is a letter of the word 'ASSASSINATION'}

Accepted Answer

NA

Question 14

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'}

Accepted Answer

NA

Question 15

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 }.

Accepted Answer

(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.

Question 16

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?

Accepted Answer

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.

Question 17

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?

Accepted Answer

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.

Question 18

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?

Accepted Answer

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.

Question 19

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?

Accepted Answer

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.

Question 20

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?

Accepted Answer

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.

Sets and Functions Questions