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  • 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 φ (B) N (C) A (D) B

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
(A) φ    (B) N    (C) A    (D) B
 

Answers (1)

The answer is the option (b).

Given that A ={1,3,5,7,9,11,13,15,17}
B = {2,4,6………..,18}
U=N={1,2,3,4,5,……..}

\begin{aligned}A'\cup (A\cup B)\cap B' &= A'\cup [(A\cap B')\cup (B\cap B')] \\ & = A'\cup (A\cap B')\cup \phi&[A\cap A' = \phi] \\ &= A'\cup(A\cap B') \\ & = (A'\cup A) \cap (A'\cup B') \\ &=N\cup (A'\cup B')\\& = A'\cup B' \\ & = (A\cap B)' = \phi ' = N\end{aligned}

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