Question

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {2, 4, ..., 18} and N the set of natural numbers is the universal set, then A′ ∪ (A ∪ B) ∩ B′) is ______.

Options

• Φ

• N

• A

• B

Answer 1

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {2, 4, ..., 18} and N the set of natural numbers is the universal set, then A′ ∪ (A ∪ B) ∩ B′) is N.

Explanation:

Given that, A = {1, 3, 5, 7, 9, 11, 13, 15, 17}

B = {2, 4, ..., 18}

U = N = {1, 2, 3, 4, 5, ...}

A' ∪ (A ∪ B) ∩ B' = A' ∪ [(A ∩ B') ∪ (B ∩ B')]

= A' ∪ (A ∪ B') ∩ Φ   ......[∵ A ∩ A' = Φ]

= A' ∪ (A ∩ B')

= (A' ∪ A) ∩ (A' ∪ B')

= N ∩ (A' ∪ B')   ......[∵ A' ∪ A = N]

= A' ∪ B'

= (A ∩ B)'

= (Φ)'

= N   ......[∵ A ∩ B = Φ]