# 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.