Exams › JEE Advanced › Maths

Let A, B, C, D be subsets of a universal set X. Which of the following statements are necessarily true? (P) (C ∪ B) (A ∩ (B C)) = B^c ∩ C^c (Q) A ∪ [(A ∪ B) ∩ (B ∪ (A ∩ B))] = A ∪ B (R) C^c ∪ [D^c ∪ (D C^c)] = C^c ∪ D^c (S) [(A ∪ B) ∪ B^c] ∩ [B^c ∩ (A ∪ B)] = A B

1. (C ∪ B) (A ∩ (B C)) = B^c ∩ C^c
2. A ∪ [(A ∪ B) ∩ (B ∪ (A ∩ B))] = A ∪ B
3. C^c ∪ [D^c ∪ (D C^c)] = C^c ∪ D^c
4. [(A ∪ B) ∪ B^c] ∩ [B^c ∩ (A ∪ B)] = A B

Correct answer: [(A ∪ B) ∪ B^c] ∩ [B^c ∩ (A ∪ B)] = A B

Solution

Only option (S) is true. (P) is false since LHS includes elements of B or C not in A's intersection. (Q) simplifies correctly to A ∪ B (true). (R) fails because D C^c = D ∩ C, and C^c ∪ D^c ∪ (D ∩ C) = C^c ∪ D^c ∪ D = C^c ∪ X = X, not C^c ∪ D^c. (S): LHS = X ∩ [B^c ∩ (A ∪ B)] = B^c ∩ (A ∪ B) = A B.

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