Question

Let $A$ and $B$ be two sets then $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)$

• A. $A\text{'}$
• B. $A$
• C. $B\text{'}$
• D. None of these

Answer 1

The correct option is A

$A\text{'}$

Step 1; Simplify $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)$

Let $X=\left\{1,2,3,4,5\right\}$

$$\begin{gathered} A=\left\{1,2\right\} \\ B=\left\{1,4,5\right\} \end{gathered}$$

Find $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)$

$A\cup B=\left\{1,2,4,5\right\}$

$$\begin{gathered} \left(A\cup B\right)\text{'}=X-\left(A\cup B\right) \\ =\left\{3\right\} \end{gathered}$$

$$\begin{gathered} A\text{'}=X-A \\ =\left\{3,4,5\right\} \end{gathered}$$ where $A-B=\left\{x/x\in Aandx\notin B\right\}$

$A\text{'}\cap B=\left\{4,5\right\}$ where $A\cap B=\left\{x/x\in Aandx\in B\right\}$

$\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)=\left\{3,4,5\right\}$

Step 2: Checking the given options to obtain the correct option

Option (A):

To check if $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)=A\text{'}$

$A\text{'}=\left\{3,4,5\right\}$

$$\begin{gathered} \Rightarrow LHS=\left\{3,4,5\right\} \\ \Rightarrow RHS=\left\{3,4,5\right\} \\ \Rightarrow RHS=LHS \end{gathered}$$

Therefore, option (A) is correct.

Option (B):

To check if $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)=A$

$A=\left\{1,2\right\}$

$$\begin{gathered} \Rightarrow LHS=\left\{3,4,5\right\} \\ \Rightarrow RHS=\left\{1,2\right\} \\ \Rightarrow RHS\ne LHS \end{gathered}$$

Therefore, option (B) is incorrect.

Option (C):

To check if $\left(A\cup B\right)\text{'}\cup \left(A\text{'}\cap B\right)=B\text{'}$

$B\text{'}=\left\{2,3\right\}$

$$\begin{gathered} \Rightarrow LHS=\left\{3,4,5\right\} \\ \Rightarrow RHS=\left\{2,3\right\} \\ \Rightarrow RHS\ne LHS \end{gathered}$$

Therefore, option (C) is incorrect.

Hence, option (A) is the correct answer.