Question

If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) ^C ∩ Z is ______.

Options

• 255

• 256

• 257

• 258

Answer 1

If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) ^C ∩ Z is 256.

Explanation:

A = {x ∈ R : |x – 2| > 1} = (–∞, 1) ∪ (3, ∞)

B = `{x ∈ R: sqrt(x^2 - 3) > 1}` = (–∞, –2) ∪ (2, ∞)

C = {x ∈ R : |x – 4| ≥ 2} = (–∞, 2] ∪ [6, ∞)

So (A ∩ B ∩ C) = (–∞, –2) ∪ [6, ∞)

Z ∩ (A ∩ B ∩ C) ^C = {–2, –1, 0, 1, 2, 3, 4, 5}

Hence no. of its subsets = 2⁸ = 256.