Question

If $A=\{1,3,5,7,9,11,13,15,17\}$, $B=\{2,4,...18\}$and $N=\{1,2,3,4......18\}$ is the universal set then $A\text{'}\cup \left(\right(A\cup B)\cap B\text{'})$ is

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

Answer 1

The correct option is D

$N$

Explanation for correct option:

Step1. Given,

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

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

$N=\{1,2,3,4,5,6.......16,17,18\}$

$\begin{array}{rcl}{A}^{\text{'}}& =& N-A=\{2,4,6,...18\}=B\\ B^{\text{'}}& =& N-B=\{1,3,5,...17\}=A\\ N& =& A\cup B=\{1,2,3,4,....17,18\}\\ N\cap B\text{'}& =& \{1,3,5,...17\}=A\end{array}$

Since, $N$ is universal set then $A\cup B=N$

Step2. Finding value for $\mathit{A}\mathbf{́}\mathbf{\cup }\mathbf{\left(}\mathbf{\right(}\mathit{A}\mathbf{\cup }\mathit{B}\mathbf{)}\mathbf{\cap }\mathit{B}\mathbf{́}\mathbf{)}\mathbf{:}$

Now,

$\begin{array}{rcl}& & Á\cup \left(\right(A\cup B)\cap B́)\\ & =& Á\cup (N\cap B́)\{A\cup B=N\}\\ & =& A\text{'}\cup A\{N\cap B\text{'}=A\}\\ & =& B\cup A\{A\text{'}=B\}\\ & =& N\end{array}$

Hence, The correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}\mathbf{.}$