Selina Concise Class 8 Math Chapter 6 Sets Exercise 6B Solutions
EXERCISE 6B
(1) Find the cardinal number of the following sets:
(i) n(A1) = 5
(ii) x = 3, 4, 5, 6
n(A2) = 4
(iii) 2p – 3 = 7
⇒ 2p = 7 + 3
⇒ 2p = 10
⇒ p = 5
Or, 2p – 3 = 6
⇒ 2p = 6 + 3
⇒ p = 4.5
Or, 2p – 3 = 5
⇒ 2P = 5+3
⇒ p = 4
Or, 2p – 3 = 4
⇒ 2p = 7
⇒ p = 3.5
Then, p = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5.
∴ n(A3) = 10
(iv) 3b – 1 = – 6
⇒ 3b = – 6 + 1
⇒ b = – 5/3
Or, 3b – 1 = – 5
⇒ 3b = – 5 + 1
⇒ b = – 4/3
Then, b = – 5/3, – 4/3, – 1, -2/3, -1/3, 0, 1, 2.
∴ n (A4) = 8
(2) p = {P, E, R, M, A, N, T}
∴ n (P) = 7
(3) State, which of the following sets are finite and which are infinite:
(i) x = ………. ,0, 1, 2, 3, 4, 5, 6, 7, 8, 9; this is infinite
(ii) Finite
5x – 3 ≤ 20
⇒ 5x – 3 + 3 ≤ 20 + 3
⇒ 5x ≤ 23
⇒ x ≤ 23/5
⇒ x ≤ 4.6
Then, x = {0, 2, 3, 4}
(iii) Infinite
When, x = 6 y = 3 × 6 – 2 = 18 – 2 = 16
When, x = 7 y = 3 × 7 – 2 = 21 – 2 = 19
When, x = 8 y = 3 × 8 – 2 = 24 – 2 = 22
∴ P = {16, 19, 22, …..}
(iv) Finite
Here, r = 3/n
When, n = 7 r = 3/7
When, n = 8 r = 3/8
When, n = 9 r = 3/9 = 1/3
When, n = 10 r = 3/10
When, n = 11 r = 3/11
……
When, n = 15 r = 3/15 = 1/5
Then, M = {3/7, 3/8, 1/ 3,……..1/5}
(4) Find, which of the following sets are singleton sets:
(i) Singleton set
(ii) 7x – 3 = 11
⇒ 7x = 11 + 3 = 14
⇒ x = 14/7
⇒ x = 2
Therefore, it is a singleton set.
(iii) 2y + 1 < 3
⇒ 2y + 1 – 1 < 3 – 1
⇒ 2y = 2
⇒ y = 1
∴ B = { 0 }
Therefore, it is a singleton set.
(5) Find, which of the following sets are empty:
(i) “The set of points of intersection of two parallel lines” is an empty set because two parallel lines do not intersect anywhere.
(ii) x = 6
∴ A = {6}
Therefore, it is not an empty set.
(iii) x2 + 4 = 0
⇒ x2 = – 4
⇒ x = √(- 4) which is not a natural number.
But x ∈ N
∴ B = { }
Therefore, it is an empty set.
(iv) C = {8}
Therefore, it is an empty set.
(v) D = {0}
Therefore, it is an empty set.
(6) (i) A = {4, 5, 6}
x2 – 5 x – 6 = 0
⇒ x2 – 6x + x – 6 = 0
⇒ x (x – 6) + 1 (x – 6) = 0
⇒ (x + 1) (x – 6) = 0
Either, x + 1 = 0
⇒ x = – 1
Or, x – 6 = 0
⇒ x = 6
∴ B = {6, – 1}
Hence, set A and set B are not disjoint because these set have an element 6 in common.
(ii) A = {b, c, d, e}
∴ B = {m, a, s, t, e, r}
Hence, set A and set B are not joint because these set have an element e in common.
(7) State, whether the following pairs of sets are equivalent or not:
(i) 11 ≥ 2x – 1
⇒ 11 + 1 ≥ 2x – 1 + 1
⇒ 12 ≥ 2x
⇒ 6 ≥ x
∴ A = {1, 2, 3, 4, 5, 6}
∴ n (A) = 6
B = 3 ≤ y ≤ 9
∴ B = {3, 4, 5, 6, 7, 8, 9}
∴ n (B) = 7
Therefore, A and B are not equivalent; because their cardinal numbers are not equal.
(ii) Not equivalent, as the two sets are not finite.
(iii) Not, equivalent as the two sets are not finite.
(iv) P = {5, 6, 7, 8}
∴ n (P) = 4
M = {0, 1, 2, 3, 4}
∴ n (M) = 5
Therefore, P and M are not equivalent; because their cardinal numbers are not equal.
(8) State, whether the following pairs of sets are equal or not:
(i) A = {2, 4, 6, 8}
∴ n (A) = 4n
= {2n : n ∈ N and n < 5}
When, n = 1, 2n = 2 × 1 = 2
When, n = 2, 2n = 2 × 2 = 4
When, n = 3, 2n = 2 × 3 = 6
When, n = 4, 2n = 2 × 4 = 8
∴ B = {2, 4, 6, 8}
∴ Sets A and b are equal.
(ii) M = {x : x ∈ W and x + 3 < 8}
∴ x + 3 < 8
⇒ x + 3 – 3 < 8 – 3
⇒ x < 5
∴ M = {0, 1, 2, 3, 4}
N = {y : y = 2n – 1, n ∈ N and n < 5}
∴ y = 2n – 1
When, n = 1 y = 2 × 1 – 1 = 2 – 1 = 1
When, n = 2 y = 2 × 2 – 1 = 4 -1 = 3
When, n = 3 y = 2 × 3 – 1 = 6 – 1 = 5
When, n = 4 y = 2 × 4 – 1 = 8 – 1 = 7
∴ N = {1, 3, 5, 7}
∴ Set M and N are not equal.
(iii) E = {x : x2 + 8x – 9 = 0}
∴ x2 + 8x – 9 = 0
⇒ x2 + 9x – x – 9 = 0
⇒ x (x + 9) – 1 (x + 9) = 0
⇒ (x – 1) (x + 9) = 0
Either, x – 1 = 0
⇒ x = 1
Or, x + 9 = 0
⇒ x = – 9
∴ E = {1, – 9}
F = {1, – 9}
∴ Sets E and F are equal.
(iv) A = {x : x ∈ N, x < 3}
∴ A = {1, 2}
B = {y : y2 – 3y + 2 = 0}
∴ y2 – 3y + 2 = 0
⇒ y2 – 2y – y + 2 = 0
⇒ y (y – 2) – 1 (y – 2) = 0
⇒ (y – 1) (y – 2) = 0
Either, y – 1 = 0
⇒ y = 1
Or, y – 2 = 0
⇒ y = 2
∴ B = {1, 2}
∴ Sets A and B are equal.
(9) State whether each of the following sets is a finite set or an infinite set:
(i) {8, 16, 32, ……} = Infinite set
(ii) {9, 8, 7, ……., – 1 , – 2 , ……} = Infinite set
(iii) (11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0} = Finite set
(iv) {x : x 3n – 2, n ∈ W, n ≤ 8}
Substituting the value of n = {0, 1, 2, 3, 4, 5, 6, 7, 8} we get,
N = {-2, 1, 4, 7, 10, 13, 16, 19, 22} = Finite set
(v) {x : x = 3n – 2, n ∈ Z, n ≤ 8}
= {22, 19, 16, 13, 10, 7, 4, 1, -2, -5, …….} = Infinite set.
(vi) {x : x = (n + 2)/(n – 2), n ∈ W}
= {- 2, – ½, 0, ¼, 2/5, ………} = Infinite set.
(10) Answer, whether the following statements are true or false. Give reasons.
(i) True, since both the sets have 10 elements.
(ii) False, since E = {1, 2, 4, 8, 16} and F = {1, 2, 4, 5, 10, 20}
(iii) False, since A = {19, 18, ……., 0, – 1, – 2, …….}
(iv) False, since A = {2}
(v) False, since the given set has 3, 5, 7, 11, etc
(vi) False
(vii) True
(viii) False, the sets are equivalent.
