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GSEB Solutions for Class 8 Mathematics – Introduction to Set

GSEB Solutions for Class 8 Mathematics – Introduction to Set (English Medium)

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Exercise

Solution 1:

  1. 6 ϵ {1, 2, 4, 6}6 lies in the set {1, 2, 4, 6}
  2. {20}⊂{20, 30, 40}Since, 20 lies in {20, 30, 40}
  3. 7 ϵ {x/x is a prime natural number}7 is a prime natural number.
  4. 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
  5. 1, 2, 3 are natural numbers.
  6. {-1, 1, 0}⊄NIn the set {-1, 1, 0}, -1 is not a natural number.
  7. If A = {a, b, c} and B = {1, 2, 3}, then A ~ B.n(A) = 3 and n(B) = 3.

Solution 2:
gseb-solutions-for-class-8-mathematics-introduction-to-set-1

Solution 3:

  1. {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.
  2. {5}Since the given set {5} has only one element, it is a singleton set.
  3. {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.
  4. {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:

  1. The given set is a finite set.Reason:
    The number of citizens of India is a finite positive integer at a given time.
  2. The given set is a finite set.Reason:
    The number of three digit numbers greater than 100 is a finite positive integer, 899.
  3. The given set is an infinite set.Reason:
    There are infinite numbers like 7, 17, 27, … having 7 at the units place.
  4. The given set is an infinite set.Reason: There are infinite prime numbers.

Solution 5:

  1. P ~ QHere, n(p) = 3 and n(Q) = 3, but P and Q do not have identical elements.
    Hence, they are not equal sets.
  2. 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
  3. 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
  4. Sets D and E have the same elements.
    ∴D = E and D ~ E
  5. 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 one-to-one correspondences between A = {x, y} and B = {a, b}.
gseb-solutions-for-class-8-mathematics-introduction-to-set-2

Solution 8:

  1. A ∪ B = {x, y, z, w, a, b, c} and A ∩ B = {x, y}Venn diagram is as follows:
  2. S ∪ R = {5, 10, 15, 25, 20} and S ∩ R = {10, 15}

    Venn diagram is as follows:

    gseb-solutions-for-class-8-mathematics-introduction-to-set-3

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}.

  1. A ∩ B= {4, 7}
  2. = {0, 1, 4, 5, 7} ∩ {3, 4, 6, 7}
  3. (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}
  4. = {0, 1, 2, 3, 4, 5, 6, 7}
  5. Now, (A ∪ B) ∪ (B ∪ C)
  6. (B ∪ C) = {3, 4, 6, 7} ∪ {2, 5, 6, 7}
  7. (A ∪ B) = {0, 1, 4, 5, 7} ∪ {3, 4, 6, 7}
  8. A ∩ (B ∪ C)B ∪ C = {3, 4, 6, 7} ∪ {2, 5, 6, 7}Now, A ∩ (B ∪ C)= {4, 5, 7}
  9. = {0, 1, 4, 5, 7} ∩ {2, 3, 4, 5, 6, 7}
  10. = {2, 3, 4, 5, 6, 7}
  11. A = {0, 1, 4, 5, 7}
  12. (A ∪ C) ∩ B= {0, 1, 2, 4, 5, 6, 7}Now, (A ∪ C) ∩ B= {4, 6, 7}
  13. = {0, 1, 2, 4, 5, 6, 7} ∩ {3, 4, 6, 7}
  14. B = {3, 4, 6, 7}
  15. A ∪ C = {0, 1, 4, 5, 7} ∪ {2, 5, 6, 7}
  16. U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Practice 1

Solution 1:

  1. 3  ϵ {1, 2, 3, 4} Since 3 lies in the set {1, 2, 3, 4}
  2. 100  ∉ {1, 2, 3, ….., 99}Since 100 does not lie in the set {1, 2, 3, …., 99}
  3. 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.
  4. 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.
  5. 0  ∉ {x/x is a natural number}0 is not a natural number.

Solution 2:
gseb-solutions-for-class-8-mathematics-introduction-to-set-4

Solution 3:

  1. Singleton sets in the given table: {Dang}, {Kachchh}
  2. All the sets in the given table are finite sets. An empty set is also a finite set.
  3. 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:

  1. N⊂Z
    ‘N’ denotes Natural numbers and ‘Z’ denotes Integers.
    Since natural numbers are positive integers, set N is subset of set Z.
  1. {3, 1, -1}⊄N
    -1 is not a natural number.
    ∴ -1 ⊄ N
    ∴ {3, 1, -1} ⊄ N
  1. Z⊂Q
    ‘Z’ denotes Integers and ‘Q’ denotes Rational numbers.
    Since, group of integers are Rational numbers, Z ⊂ Q

gseb-solutions-for-class-8-mathematics-introduction-to-set-5

Solution 2:
gseb-solutions-for-class-8-mathematics-introduction-to-set-6

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:

  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}
  2. 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}
  3. 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}

  1. A ⋃ B= {0, 1, 2, 3, 4, 5, 8, 9}
  2. = {0, 1, 2, 3, 5} ⋃ {2, 3, 4, 8, 9}
  3. A ⋂ B= {2, 3}
  4. = {0, 1, 2, 3, 5} ⋂ {2, 3, 4, 8, 9}
  5. For (A ⋂ C) ⋃ B,(A ⋂ C) ⋃ B = {3, 5} ⋃ {2, 3, 4, 8, 9}
  6. ∴= {2, 3, 4, 5, 8, 9}
  7. A ⋂ C = {0, 1, 2, 3, 5} ⋂ {3, 4, 5, 6, 7} = {3, 5}
  8. For (A ⋃ C) ⋃ B,= {0, 1, 2, 3, 4, 5, 6, 7}∴= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
  9. (A ⋃ C) ⋃ B = {0, 1, 2, 3, 4, 5, 6, 7} ⋃ {2, 3, 4, 8, 9}
  10. A ⋃ C = {0, 1, 2, 3, 5} ⋃ {3, 4, 5, 6, 7}
  11.  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}
gseb-solutions-for-class-8-mathematics-introduction-to-set-7