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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
Solution
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]
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