Question

Let $A=\left\{1,2,3,4\right\},B=\left\{2,4,6\right\}$. Then the number of sets $C$ such that $A\cap B\subseteq C\subseteq A\cup B$ is

• A. $6$
• B. $9$
• C. $8$
• D. $10$

Answer 1

The correct option is C

$8$

Determine the number of sets $C$ such that $A\cap B\subseteq C\subseteq A\cup B$

Given $A=\left\{1,2,3,4\right\}$ and $B=\left\{2,4,6\right\}$

Also $A\cap B\subseteq C\subseteq A\cup B$

Now, $A\cap B=\left\{2,4\right\}$ and $A\cup B=\left\{1,2,3,4,6\right\}$

Therefore we have,

$$\begin{gathered} A\cap B\subseteq C\subseteq A\cup B \\ \Rightarrow \left\{2,4\right\}\subseteq C\subseteq \left\{1,2,3,4,6\right\} \end{gathered}$$

Hence the possible number of sets $C$ which satisfy the given condition are:

$C=\left\{(2,4),(1,2,3,4,6),(2,4,1),(2,4,3),(2,4,6),(2,4,1,3),(2,4,1,6),(2,4,3,6)\right\}$

Therefore, the number of sets $C=8$

Hence, option C is the correct answer.