ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 6 Operation on sets Venn Diagrams Ex 6.2
Question 1.
From the adjoining Venn diagram, find the following sets :
(i) A
(ii) B
(iii) ξ
(iv) A’
(v) B’
(vi) A ∪ B
(vii) A ∩ B
(viii) (A ∪ B)’
(ix) (A ∩ B)’
Solution:
From the given Venn diagram, we find that:
(i) A = {0, 7, 9, 11, 5, 8}
(ii) B = {2, 6, 5, 8}
(iii) ξ = {0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 12}
(iv) A’= {1,2, 4, 6, 12}
(v) B’ = {0, 1,4, 7, 9, 11, 12}
(vi) A ∪ B = {0, 9, 7, 11, 5, 8, 2, 6}
(vii) A ∩ B = {5, 8}
(viii) (A ∪ B)’ = {1, 4, 12}
(ix) (A ∩ B)’ = {0, 1,2, 4, 6, 7, 9, 11, 12}
Question 2.
From the adjoining Venn diagram, find the following sets :
(i) P
(ii) Q
(iii) ξ
(iv) P’
(v) Q’
(vi) P ∪ Q
(vii) P ∩ Q
(viii) (P ∪ Q)’
(ix) (P ∩ Q)’
Solution:
From the given Venn diagram, we find that
(i) P = {a, b, d, e, f g, h, i}
(ii) Q = {b, d, e}
(iii) ξ = {a, b, c, d, e, f g, h, i, j)
(iv) P’ = {c, j}
(v) Q’ = {a, c, f, g, h, i, j}
(vi) P ∪ Q = {a, b, d, e, f g, h, i}
(vii) P ∩ Q = {b, d, e}
(viii) (P ∪ Q)’ = {c, j}
(ix) (P ∩ Q)’ = {a, c, f g, h, i, j}
Question 3.
From the adjoining Venn diagram, find the following sets :
(i) ξ
(ii) A ∩ B
(iii) A ∩ B ∩ C
(iv) C’
(v) A – C
(vi) B – C
(vii) C – B
(viii) (A ∪ B)’
(ix) (A ∪ B ∪ C)’
Solution:
(i) ξ = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
(ii) A ∩ B = {8,0, 5}
(iii) A ∩B ∩C = {0, 5}
(iv) C’ = {2, 7, 8, 9, 10, 11, 12}
(v) A – C = (8, 10}
(vi) B – C = {7, 8, 11}
(vii) C – B = {3, 4, 6}
(viii) (A ∪ B)’ = {2, 4, 6, 9, 12}
(ix) (A ∪B ∪ )’ = {2, 9, 12}
Question 4.
Draw Venn diagrams to show the relationship between the following pairs of sets :
(i) A = {x | x ϵ N, x = 2n, n ≤ 5} and
B = {x | x ϵ W, x = 4n, n < 5}
(ii) A = {prime factors of 42} and
B = {prime factors of 60}
(iii) P = {x | x ϵ W, x < 10} and
Q = {prime factors of 210}
Solution:
(i) Here, A = {2, 4, 6, 8, 10}
B = {0, 4, 8, 12, 16}
∴ A ∩ B = {4, 8}
So the sets A and B are overlapping
The relationship betwen the sets A and B is represented
by the adjoining Venn digaram.
(ii) Here, A = {2, 3, 7}
B = {2, 3, 5}
∴ A ∩ B = {2, 3}
So the sets A and B are overlapping.
The relationship between the sets A and B is represented
by the adjoining Venn diagram.
(iii) Here, P = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Q = {2, 3, 5, 6, 7} .
∴ P ∩ Q = {2, 3, 5, 6, 7}. So Q ⊂ P.
The relationship between the sets P and Q is represented
by the adjoining Venn diagram.
Question 5.
Draw a Venn diagram to illustrate the following information:
n (A) = 22, n (B) = 18 and n (A ∩ B) = 5.
Hence find:
(i) n (A ∩ B)
(ii) n (A – B)
(iii) n (B – A)
Solution:
n (only A) = n (A) – n (A ∩ B)
= 22 – 5 = 17
n (only B) = n (B) – n (A ∩ B)
= 18 – 5 = 13
(i) n (A ∪ B) = 35
(ii) n (A – B) = 17
(iii) (B – A) = 13
Question 6.
Draw a Venn diagram to illustrate the following information: n (A) = 25, n (B) = 16, n (A ∩ B) = 6 and n ((A ∪ B)’) = 5
Hence find:
(i) n (A ∪ B)
(ii) n (ξ)
(iii) n (A – B)
(iv) n (B – A)
Solution:
n (only A) = n (A) – n (A ∩ B)
= 25 – 6= 19
n (only B) = n (B) – n(A∩B)
= 16 – 6 = 10
n((A ∪ B)’) = 5
(i) n(A ∪ B) = 35
(ii) n(ξ) = 40
(iii) n(A – B) = 19
(iv) n(B – A) = 10
Question 7.
Given n(ξ) = 25, n (A’) = 7, n (B) = 10 and B ⊂ A. Draw a Venn diagram to illustrate this information. Hence, find the cardinal number of the set A – B.
Solution:
Given n (ξ) = 25
n (A’) = 7
n (B) = 10
and B ⊂ A
n (A) = n (ξ) – n (A’)
= 25 – 7 = 18
The cardinal number of the set A – B = 8.
Question 8.
In a group of 50 boys, 20 play only cricket, 12 play only football and 5 boys play both the games. Draw a Venn diagram and find the number of boys who play
(i) atleast one of the two games cricket or football.
(ii) neither cricket nor football.
Solution:
Let ξ be all the boys.
Let A be the students who play cricket
and B be the students who play football.
Then, n (ξ) = 50
n (only A) = 20
n (only B) = 12
n (A ∩ B) = 5
(i) at least one of the two games cricket or football
= n (A ∪ B) = 20 + 5 + 12 = 37 neither cricket nor football =13.
Question 9.
In a group of 40 students, 26 students like orange but not a banana, while 32 students like orange. If all the students like atleast one of the two fruits, find the number of students who like
(i) both orange and banana
(ii) only banana.
Draw a Venn diagram to represent the data.
Solution:
ξ = Group of total students
A = Those students, who like orange
B = Those students, who like banana
∴ n(ξ) = 40
n (A – B) = 26
n (B) = 32
n(A ∪ B) = 40
(i) Number of those students,
who like both orange and banana i.e. n (A ∩ B) = 6.
(ii) Number of those students
who like only banana = 40 – 32 = 8.
Question 10.
In a group of 60 persons, 45 speak Bengali, 28 speak English and all the persons speak at least one language. Find how many people speak both Bengali and English. Draw a Venn diagram.
Solution:
Let ξ = group of persons
A = Those persons who speak Bengali
B = Those persons who speak English
n (ξ) = 60
n (A) = 45
n (B) = 28
and n (A ∪ B) = 60
n(A ∩ B) = n (A) + n (B) – n(A ∪ B)
= 45 + 28 – 60 = 13.
