Question

Q. 5: Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.

Find A′, B′, A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′∩ B′.

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Answer 1

Given,

U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}

A′ = {1, 4, 5, 6}

B′ = { 1, 2, 6 }.

Hence, A′ ∩ B′ = { 1, 6 }

Also A ∪ B = { 2, 3, 4, 5 }

(A ∪ B)′ = { 1, 6 }

Therefore, ( A ∪ B )′ = { 1, 6 } = A′ ∩ B′

Answer 2

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(i) For X = {1, 2, 3, 4, 5, 6}, it is given that n ∈ X, but 2n ∉ X.

Let, A = {x | x ∈ X and 2x ∉ X}

Now, 1 ∉ A as 2.1 = 2 ∈ X

2 ∉ A as 2.2 = 4 ∈ X

3 ∉ A as 2.3 = 6 ∈ X

But 4 ∈ A as 2.4 = 8 ∉ X

5 ∈ A as 2.5 = 10 ∉ X

6 ∈ A as 2.6 = 12 ∉ X

Therefore, A = {4, 5, 6}

(ii) Let B = {x | x ∈ X and x + 5 = 8}

Here, B = {3} as x = 3 ∈ X and 3 + 5 = 8 and there is no other element belonging to X such that x + 5 = 8.

(iii) Let C = {x | x ∈ X, x > 4}

Therefore, C = {5, 6}

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Hope it's Helpful.....:)