Question

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 $(A^{\prime }\cup (A\cup B)\cap B^{\prime })$ is

• A. $A$
• B. $\varphi$
• C. $B$
• D. $N$

Answer 1

The correct option is D $N$

$A=\{1,3,5,7,9,11,13,15,17\},$

$B=\{2,4,...,18\}$

$N$ is the set of natural numbers, which is the universal set

$A^{\prime }\cup (A\cup B)\cap B^{\prime }=A^{\prime }\cup \left[\right(A\cap B^{\prime })\cup (B\cap B^{\prime }\left)\right]$

$\{\because (A\cup B)\cap C=(A\cap C)\cup (B\cap C\left)\right\}$

$\Rightarrow A^{\prime }\cup (A\cup B)\cap B^{\prime }=A^{\prime }\cup \left[\right(A\cap B^{\prime })\cup \varphi$

$\{\because A\cap A^{\prime }=\varphi \}$

$=A^{\prime }\cup (A\cap B^{\prime })$

$=(A^{\prime }\cup A)\cap (A^{\prime }\cup B^{\prime })$

$\{\because A\cup (B\cap C)=(A\cup B)\cap (A\cup C\left)\right\}$

$\Rightarrow A^{\prime }\cup (A\cup B)\cap B^{\prime }=(A^{\prime }\cup A)\cap (A^{\prime }\cup B^{\prime })$

$\Rightarrow A^{\prime }\cup (A\cup B)\cap B^{\prime }=N\cap (A^{\prime }\cup B^{\prime })$

$\{\because A^{\prime }\cup A=N\}$

$\Rightarrow A^{\prime }\cup (A\cup B)\cap B^{\prime }=A^{\prime }\cup B^{\prime }$

$=(A\cap B{)}^{\prime }$

$={\varphi }^{\prime }$

$=N$

Hence, option (B) is correct.