RD Chapter 1 Sets Ex 1.8 Solutions
Question - 1 : - If A and B are two sets such that n (A тИк B) = 50, n (A) = 28 and n (B) = 32, find n (A тИй B).
Answer - 1 : -
We have,
n (A тИк B) = 50
n (A) = 28
n (B) = 32
We know, n (A тИк B) = n (A) + n (B) тАУ n (A тИй B)
Substituting the values we get
50 = 28 + 32 тАУ n (A тИй B)
50 = 60 тАУ n (A тИй B)
тАУ10 = тАУ n (A тИй B)
тИ┤ n (A тИй B) = 10
Question - 2 : - If P and Q are two sets such that P has 40 elements, P тИк Q has 60 elements and P тИй Q has 10 elements, how many elements does Q have?
Answer - 2 : -
We have,
n (P) = 40
n (P тИк Q) = 60
n (P тИй Q) =10
We know, n (P тИк Q) = n (P) + n (Q) тАУ n (P тИй Q)
Substituting the values we get
60 = 40 + n (Q)тАУ10
60 = 30 + n (Q)
N (Q) = 30
тИ┤ Q has 30 elements.
Question - 3 : - In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?
Answer - 3 : -
We have,
Teachers teaching physics or math = 20
Teachers teaching physics and math = 4
Teachers teaching maths = 12
Let teachers who teach physics be тАШn (P)тАЩ and for Maths be тАШn (M)тАЩ
Now,
20 teachers who teach physics or math = n (P тИк M) = 20
4 teachers who teach physics and math = n (P тИй M) = 4
12 teachers who teach maths = n (M) = 12
We know,
n (P тИк M) = n (M) + n (P) тАУ n (P тИй M)
Substituting the values we get,
20 = 12 + n (P) тАУ 4
20 = 8 + n (P)
n (P) =12
тИ┤ There are 12 physics teachers.
Question - 4 : - 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 like both coffee and tea?
Answer - 4 : -
We have,
A total number of people = 70
Number of people who like Coffee = n (C) = 37
Number of people who like Tea = n (T) = 52
Total number = n (C тИк T) = 70
Person who likes both would be n (C тИй T)
We know,
n (C тИк T) = n (C) + n (T) тАУ n (C тИй T)
Substituting the values we get
70 = 37 + 52 тАУ n (C тИй T)
70 = 89 тАУ n (C тИй T)
n (C тИй T) =19
тИ┤ There are 19 persons who like both coffee and tea.
Question - 5 : - Let A and B be two sets such that: n (A) = 20, n (A тИк B) = 42 and n (A тИй B) = 4. Find
(i) n (B)
(ii) n (A тАУ B)
(iii) n (B тАУ A)
Answer - 5 : -
(i) n (B)
We know,
n (A тИк B) = n (A) + n (B) тАУ n (A тИй B)
Substituting the values we get
42 = 20 + n (B) тАУ 4
42 = 16 + n (B)
n (B) = 26
тИ┤ n (B) = 26
(ii) n (A тАУ B)
We know,
n (A тАУ B) = n (A тИк B) тАУ n (B)
Substituting the values we get
n (A тАУ B) = 42 тАУ 26
= 16
тИ┤ n (A тАУ B) = 16
(iii) n (B тАУ A)
We know,
n (B тАУ A) = n (B) тАУ n (A тИй B)
Substituting the values we get
n (B тАУ A) = 26 тАУ 4
= 22
тИ┤ n (B тАУ A) = 22
Question - 6 : - A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?
Answer - 6 : -
We have,
People who like oranges = 76%
People who like bananas = 62%
Let people who like oranges be n (O)
Let people who like bananas be n (B)
Total number of people who like oranges or bananas = n (O тИк B) = 100
People who like both oranges and bananas = n (O тИй B)
We know,
n (O тИк B) = n (O) + n (B) тАУ n (O тИй B)
Substituting the values we get
100 = 76 + 62 тАУ n (O тИй B)
100 = 138 тАУ n (O тИй B)
n (O тИй B) = 38
тИ┤ People who like both oranges and banana is 38%.
Question - 7 : - In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
(i) How many can speak both Hindi and English.
(ii) How many can speak Hindi only.
(iii) how many can speak English only.
Answer - 7 : -
We have,
Let, total number of people be n (P) = 950
People who can speak English n (E) = 460
People who can speak Hindi n (H) = 750
(i) How many can speak both Hindi and English.
People who can speak both Hindi and English = n (H тИй E)
We know,
n (P) = n (E) + n (H) тАУ n (H тИй E)
Substituting the values we get
950 = 460 + 750 тАУ n (H тИй E)
950 = 1210 тАУ n (H тИй E)
n (H тИй E) = 260
тИ┤ Number of people who can speak both English and Hindi are 260.
(ii) How many can speak Hindi only.
We can see that H is disjoint union of n (HтАУE) and n (H тИй E).
(If A and B are disjoint then n (A тИк B) = n (A) + n (B))
тИ┤ H = n (HтАУE) тИк n (H тИй E)
n (H) = n (HтАУE) + n (H тИй E)
750 = n (H тАУ E) + 260
n (HтАУE) = 490
тИ┤ 490 people can speak only Hindi.
(iii) How many can speak English only.
We can see that E is disjoint union of n (EтАУH) and n (H тИй E)
(If A and B are disjoint then n (A тИк B) = n (A) + n (B))
тИ┤ E = n (EтАУH) тИк n (H тИй E).
n (E) = n (EтАУH) + n (H тИй E).
460 = n (H тАУ E) + 260
n (HтАУE) = 460 тАУ 260 = 200
тИ┤ 200 people can speak only English.