Complement of a Set and Its Properties

Correct answer: A. 3 (तीन)

Explanation: The set B contains the multiples of 9 from 1 to 54: B = {9, 18, 27, 36, 45, 54}. The complement A′ contains all elements of U that are not divisible by 6. Among the elements of B, 18, 36, and 54 are divisible by 6, so they are excluded from A′. The remaining elements are 9, 27, and 45. Therefore, A′ ∩ B = {9, 27, 45}, and its cardinality is n(A′ ∩ B) = 3.

Correct answer: A. ℝ \ {-2, 2}

Explanation: Solving x² = 4 over the real numbers gives x = −2 or x = 2. Thus A = {−2, 2}. The complement is always determined relative to the stated universal set. Since U = ℝ, remove these two elements from all real numbers: A' = ℝ \ {−2, 2}. Option B is A itself, while options C and D incorrectly ignore the two excluded real numbers.

Correct answer: A. 50

Explanation: By De Morgan’s law, A' ∩ B' = (A ∪ B)'. Taking the complement once more gives (A' ∩ B')' = ((A ∪ B)')' = A ∪ B, using the double-complement law. The question already gives n(A ∪ B) = 50, so n((A' ∩ B')') = 50. The values of n(U), n(A), and n(B) are not needed after this identity is applied.

Correct answer: A. 77

Explanation: By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. In U, the multiples of 10 are 10, 20, ..., 90, so there are 9; the multiples of 18 are 18, 36, 54, 72, 90, so there are 5. Their only common element is 90 because lcm(10,18) = 90. Thus n(A ∪ B) = 9 + 5 - 1 = 13, and n(A′ ∩ B′) = 90 - 13 = 77. Therefore, option A is correct.