GSEB Solutions for Class 8 Mathematics – Introduction to Set (English Medium)
GSEB SolutionsMathsScience
Exercise
Solution 1:
 6 ϵ {1, 2, 4, 6}6 lies in the set {1, 2, 4, 6}
 {20}⊂{20, 30, 40}Since, 20 lies in {20, 30, 40}
 7 ϵ {x/x is a prime natural number}7 is a prime natural number.
 9 ∉ {x/x is multiple of 18}Multiple of 18 are 18, 36, 48 ….. Hence, 9 does not lie in that set.{1, 2, 3}⊂N
 1, 2, 3 are natural numbers.
 {1, 1, 0}⊄NIn the set {1, 1, 0}, 1 is not a natural number.
 If A = {a, b, c} and B = {1, 2, 3}, then A ~ B.n(A) = 3 and n(B) = 3.
Solution 2:
Solution 3:
 {x/x is a prime number less than 3}The only prime less than 3 is 2.
∴The given set is {2} which is a singleton set.  {5}Since the given set {5} has only one element, it is a singleton set.
 {x/x + 1 = 1, x ϵ N}x + 1 = 1∴x = 1 – 1∴x = 0But, 0 ϵ N
Hence, there is not a single element in the given set. So, the given set is an empty set.  {x/x is the additive identity} or {x/x is a neural element for addition}
The additive identity is 0.So, the given set is {0} and has only one element.
Hence, the given set is a singleton set.
Solution 4:
 The given set is a finite set.Reason:
The number of citizens of India is a finite positive integer at a given time.  The given set is a finite set.Reason:
The number of three digit numbers greater than 100 is a finite positive integer, 899.  The given set is an infinite set.Reason:
There are infinite numbers like 7, 17, 27, … having 7 at the units place.  The given set is an infinite set.Reason: There are infinite prime numbers.
Solution 5:
 P ~ QHere, n(p) = 3 and n(Q) = 3, but P and Q do not have identical elements.
Hence, they are not equal sets.  F = { } = ØThus, n(F) = 0G = {x/x is four digit number less than 1000}
There is no four digit number less than 1000.
∴G = Ø
i.e. n(G) = 0
Thus, we have F = G and F ~ G  A = {1, 4, 9, 16} andB = {x/x is a perfect square number less than 25}
∴B = {1, 4, 9, 16}∴A = B
Moreover, all the equal sets are equivalent sets.
∴A ~ B  Sets D and E have the same elements.
∴D = E and D ~ E  Here, n(A) = 3 and n(B) = 3.But those elements are different.
∴ A ~ B
Solution 6:
U = N = {1, 2, 3, …}
A = {1, 2, 3, …10}
Now, A’ = {x/x ϵ U and x ϵ A}
∴ A’ = {11, 12, 13, …}
Again, (A’)’ = {x/x ϵ U and x ϵ A’}
∴ (A’)’ = {1, 2, 3, …10} = A
Solution 7:
There are two possible onetoone correspondences between A = {x, y} and B = {a, b}.
Solution 8:
 A ∪ B = {x, y, z, w, a, b, c} and A ∩ B = {x, y}Venn diagram is as follows:

S ∪ R = {5, 10, 15, 25, 20} and S ∩ R = {10, 15}
Venn diagram is as follows:
Solution 9:
From the given Venn diagram, we have
A = {0, 1, 4, 5, 7}
B = {3, 4, 6, 7}
C = {2, 5, 6, 7}
U = {0, 1, 2, 3…9}.
 A ∩ B= {4, 7}
 = {0, 1, 4, 5, 7} ∩ {3, 4, 6, 7}
 (A ∪ B) ∪ (B ∪ C)= {0, 1, 3, 4, 5, 6, 7}
= {2, 3, 4, 5, 6, 7}= {0, 1, 3, 4, 5, 6, 7} ∪ {2, 3, 4, 5, 6, 7}  = {0, 1, 2, 3, 4, 5, 6, 7}
 Now, (A ∪ B) ∪ (B ∪ C)
 (B ∪ C) = {3, 4, 6, 7} ∪ {2, 5, 6, 7}
 (A ∪ B) = {0, 1, 4, 5, 7} ∪ {3, 4, 6, 7}
 A ∩ (B ∪ C)B ∪ C = {3, 4, 6, 7} ∪ {2, 5, 6, 7}Now, A ∩ (B ∪ C)= {4, 5, 7}
 = {0, 1, 4, 5, 7} ∩ {2, 3, 4, 5, 6, 7}
 = {2, 3, 4, 5, 6, 7}
 A = {0, 1, 4, 5, 7}
 (A ∪ C) ∩ B= {0, 1, 2, 4, 5, 6, 7}Now, (A ∪ C) ∩ B= {4, 6, 7}
 = {0, 1, 2, 4, 5, 6, 7} ∩ {3, 4, 6, 7}
 B = {3, 4, 6, 7}
 A ∪ C = {0, 1, 4, 5, 7} ∪ {2, 5, 6, 7}
 U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Practice 1
Solution 1:
 3 ϵ {1, 2, 3, 4} Since 3 lies in the set {1, 2, 3, 4}
 100 ∉ {1, 2, 3, ….., 99}Since 100 does not lie in the set {1, 2, 3, …., 99}
 5 ∉ {x/x is a multiple of 10}{x/x is a multiple of 10} = {…, 20, 10, 0, 10, 20, …}
Hence, 5 is not an element of this set.  2 ∉{x/x is a prime factor of 15}Prime factors of 15 are 3 and 5.
Thus, {x/x is a prime factor of 15} = {3, 5}.Hence, 2 is not an element of this set.  0 ∉ {x/x is a natural number}0 is not a natural number.
Solution 2:
Solution 3:
 Singleton sets in the given table: {Dang}, {Kachchh}
 All the sets in the given table are finite sets. An empty set is also a finite set.
 The set corresponding to the group of “Such a district of Gujarat whose boundary touches Uttar Pradesh” is any empty set because the boundary of not a single district of Gujarat touches Uttar Pradesh.
Practice 2
Solution 1:
 N⊂Z
‘N’ denotes Natural numbers and ‘Z’ denotes Integers.
Since natural numbers are positive integers, set N is subset of set Z.
 {3, 1, 1}⊄N
1 is not a natural number.
∴ 1 ⊄ N
∴ {3, 1, 1} ⊄ N
 Z⊂Q
‘Z’ denotes Integers and ‘Q’ denotes Rational numbers.
Since, group of integers are Rational numbers, Z ⊂ Q
Solution 2:
Solution 3:
A = {x/x is an even natural number less than 10}
Thus, A = {2, 4, 6, 8}
B = {2, 3, 4, 5}
Here, n(A) = 4 and n(B) = 4
Since n(A) = n(B) = 4, set A and set B are equivalent sets.
Symbolically, A ~ B.
Sets A and B do not have the same elements. Hence, set A and set B are not equal sets.
Practice 3
Solution 1:
 U = {x/x is name of months of English calendar}Thus,
U = {January, February, March…December}
A = {March, May, July, June}Now, A’ = {x/x ϵ U and x ∉ A}
∴A’ = {January, February, April, August, September, October, November, December}  U = {x/x is main colour of rainbow}Thus,
U = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}
R = {Violet, Red, Yellow}Now, R’ = {x/x ϵ U and x ∉ R}
∴R’ = {Indigo, Blue, Green, Orange}  U = {x ϵ N/x ≤ 9}Thus,
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 3, 5}Now, A’ = {x/x ϵ U and x ∉ A}
∴A’ = {1, 4, 6, 7, 8, 9}Again, (A’)’ = {x/x ϵ U and x ∉ A’}
∴(A’)’ = {2, 3, 5} = AB = {4, 5, 7}Now, B’ = {x/x ϵ U and x ∉ B}
∴B’ = {1, 2, 3, 6, 8, 9}
Again, (B’)’ = {x/x ϵ U and x ∉ B’}
∴ (B’)’ = {4, 5, 7} = B
Solution 2:
From the Venn diagram, we have
A = {0, 1, 2, 3, 5}
B = {2, 3, 4, 8, 9}
C = {3, 4, 5, 6, 7}
U = {0, 1, 2, …12}
 A ⋃ B= {0, 1, 2, 3, 4, 5, 8, 9}
 = {0, 1, 2, 3, 5} ⋃ {2, 3, 4, 8, 9}
 A ⋂ B= {2, 3}
 = {0, 1, 2, 3, 5} ⋂ {2, 3, 4, 8, 9}
 For (A ⋂ C) ⋃ B,(A ⋂ C) ⋃ B = {3, 5} ⋃ {2, 3, 4, 8, 9}
 ∴= {2, 3, 4, 5, 8, 9}
 A ⋂ C = {0, 1, 2, 3, 5} ⋂ {3, 4, 5, 6, 7} = {3, 5}
 For (A ⋃ C) ⋃ B,= {0, 1, 2, 3, 4, 5, 6, 7}∴= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
 (A ⋃ C) ⋃ B = {0, 1, 2, 3, 4, 5, 6, 7} ⋃ {2, 3, 4, 8, 9}
 A ⋃ C = {0, 1, 2, 3, 5} ⋃ {3, 4, 5, 6, 7}
 The Universal set, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Solution 3:
A = {x/x is a natural number less than 5}
∴ A = {1, 2, 3, 4}
B = {x/3 < x < 7; x ϵ N}
∴ B = {4, 5, 6}
Now, A ⋃ B = {1, 2, 3, 4} ⋃ {4, 5, 6}
= {1, 2, 3, 4, 5, 6}
And A ⋂ B = {1, 2, 3, 4} ⋂ {4, 5, 6}
= {4}