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
$1)\text{ }6$
$2)\text{ 9}$
$3)\text{ 8}$
$4)\text{ 10}$

Answer 1

Hint: In this question we have been given with two sets $A$ and $B$ for which we have to find the number of sets $C$ such that the condition $A\cap B\subseteq C\subseteq A\cup B$ is fulfilled. We will solve this question by first finding the intersection and union of the two sets and substituting them. Then based on the property of the subset of sets, we will find the number of sets which can be $C$ and get the required solution.

Complete step by step answer:
We have the sets given to us as:
$\Rightarrow A=\left\{ 1,2,3,4 \right\}$
$\Rightarrow B=\left\{ 2,4,6 \right\}$
We need to find the sets $C$ such that the condition $A\cap B\subseteq C\subseteq A\cup B$ is satisfied.
We have the intersection of the sets as:
$\Rightarrow A\cap B=\left\{ 2,4 \right\}$
We have the union of the sets as:
$\Rightarrow A\cap B=\left\{ 1,2,3,4,6 \right\}$
Substituting the values in the expression, we get:
$\Rightarrow \left\{ 2,4 \right\}\subseteq C\subseteq \left\{ 1,2,3,4,6 \right\}$
Now we know that the set $\left\{ 2,4 \right\}$ is a subset of $C$ and $C$ is a subset of the set $\left\{ 1,2,3,4,6 \right\}$.
This means that the elements $\left\{ 2,4 \right\}$ will be present in the set $C$ and also $\left\{ 2,4 \right\}$ is one value of $C$.
Similarly, $\left\{ 1,2,3,4,6 \right\}$is one value of $C$.
Now for the remaining sets, we have total $3$ elements left, out of which there can be one or two elements selected from $\left\{ 1,3,6 \right\}$
On selecting one element and adding it to the set $\left\{ 2,4 \right\}$, we get:
$C=\left\{ 2,4,1 \right\}$
$C=\left\{ 2,4,3 \right\}$
$C=\left\{ 2,4,6 \right\}$
On selecting two elements and adding it to the set $\left\{ 2,4 \right\}$, we get:
$C=\left\{ 2,4,1,3 \right\}$
$C=\left\{ 2,4,1,6 \right\}$
$C=\left\{ 2,4,3,6 \right\}$
Therefore, we have:
$C=\left\{ 2,4 \right\},\left\{ 1,2,3,4,6 \right\},\left\{ 2,4,1 \right\},\left\{ 2,4,3 \right\},\left\{ 2,4,6 \right\},\left\{ 2,4,1,3 \right\},\left\{ 2,4,1,6 \right\},\left\{ 2,4,3,6 \right\}$

So, the correct answer is “Option 3”.

Note: In these types of questions, the various notations of sets should be remembered. It is to be remembered that intersection means all the elements that are common in both the sets and the union event means all the elements that are present in both the sets. It is to be remembered that we had the sign $\subseteq $ and not $\subset $ otherwise there would be only $6$ sets as $C$.