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Mathematics

State whether the following pairs of sets are equal or not :

(i) A = {2, 4, 6, 8} and B = {2n : n ∈ N and n < 5}

(ii) M = {x : x ∈ W and x + 3 < 8} and N = {y : y = 2n - 1, n ∈ N and n < 5}

(iii) E = {x : x2 + 8x - 9 = 0} and F = {1, -9}

(iv) A = {x : x ∈ N, x < 3} and
B= {y : y2 - 3y + 2 = 0}

Sets

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Answer

(i) Equal

Reason

A = {2, 4, 6, 8}

B = {2n : n ∈ N and n < 5}

When n = 1

2n = 2 x 1 = 2

When n = 2

2n = 2 x 2 = 4

When n = 3

2n = 2 x 3 = 6

When n = 4

2n = 2 x 4 = 8

B = {2, 4, 6, 8}

Hence, A = B

(ii) Not equal

Reason

M = {x : x ∈ W and x + 3 < 8}

x + 3 < 8

⇒ x < 8 - 3

⇒ x < 5

M = {0, 1, 2, 3, 4}

N = {y : y = 2n - 1, n ∈ N and n < 5}

When n = 1

y = 2 x 1 - 1 = 2 - 1 = 1

When n = 2

y = 2 x 2 - 1 = 4 - 1 = 3

When n = 3

y = 2 x 3 - 1 = 6 - 1 = 5

When n = 4

y = 2 x 4 - 1 = 8 - 1 = 7

N = {1, 3, 5, 7}

Hence, M ≠ N

(iii) Equal

Reason

E = {x : x2 + 8x - 9 = 0}

x2 + 8x - 9 = 0

⇒ x2 + 9x - x - 9 = 0

⇒ x(x + 9) - 1(x + 9) = 0

⇒ (x + 9)(x - 1) = 0

⇒ (x + 9) = 0 or (x - 1) = 0

⇒ x = -9 or 1

E = {1, -9}

F = {1, -9}

Hence, E = F

(iv) Equal

Reason

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 - 2)(y - 1) = 0

⇒ (y - 2) = 0 or (y - 1) = 0

⇒ y = 2 or y = 1

⇒ y = {1, 2}

Hence, A = B

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