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Simplify A'∪((A∪B)∩B') for Disjoint Odd/Even Sets | JEE

JEE Maths question with a full step-by-step solution.

Question
If A={1,3,5,7,9,11,13,15,17}A = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}, B={2,4,…,18}B = \{2, 4, \ldots, 18\} and NN the set of natural numbers is the universal set, then A′∪((A∪B)∩B′)A' \cup \big((A \cup B) \cap B'\big) is:
Aϕ\phi
BNNcorrect
CAA
DBB
Solution
AA (odds) and BB (evens) are disjoint ⇒A∩B′=A\Rightarrow A \cap B' = A. Step 1: Expand the bracket (distributive law):
(A∪B)∩B′=(A∩B′)∪(B∩B′)=A∪ϕ=A.(A \cup B) \cap B' = (A \cap B') \cup (B \cap B') = A \cup \phi = A.
 Step 2:
A′∪A=N.A' \cup A = N.
 Correct answer: (2)
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