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Simplify X∩(X∪Y)' to the Empty Set | JEE Sets

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

Question
If XX and YY are two sets and X′X' denotes the complement of XX, then X∩(X∪Y)′X \cap (X \cup Y)' is equal to:
AXX
BYY
Cϕ\phicorrect
DX∩YX \cap Y
Solution
By De Morgan's law, (X∪Y)′=X′∩Y′(X \cup Y)' = X' \cap Y'. Hence
X∩(X∪Y)′=X∩(X′∩Y′)=(X∩X′)∩Y′=ϕ∩Y′=ϕ.X \cap (X \cup Y)' = X \cap (X' \cap Y') = (X \cap X') \cap Y' = \phi \cap Y' = \phi.
 Correct answer: (3)
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