Question

Let A = {1, 2, 3, 4}, B = {2, 4, 6}. Then the number of sets C such that A ∩ B ⊆ C ⊆ A ⋃ B is
A) 6
B) 9
C) 8
D) 10

Answer 1

Hint: In this question, first of all, we will determine the relation that is given in the question and then, we will use the \[{2^n}\] to find the number of sets. Where, \[n\]= number of the elements in sets. The remaining elements will be inside the subset C which belongs to \[A \cap B\] and \[A \cup B\]. Hence, we will get a suitable answer.

Complete step by step solution: 
First of all, find \[A \cap B\] and \[A \cup B\]. Therefore,
Two sets \[A\]and\[B\]are given respectively,
\[\begin{array}{*{20}{c}}
  { \Rightarrow A}& = &{\{ 1,2,3,4\} }
\end{array}\] and \[\begin{array}{*{20}{c}}
  B& = &{\{ 2,4,6\} }
\end{array}\]
 To find \[A \cap B\] , choose those elements from the set B which belong to the set A. In other words, we can say that the same elements of set A and set B are taken. Now, we can write
\[ \Rightarrow \begin{array}{*{20}{c}}
  {A \cap B}& = &{\{ 1,2,3,4\} \cap \{ 2,4,6\} }
\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}
  {A \cap B}& = &{\{ 2,4\} }
\end{array}\]
And to find the \[A \cup B\], take all the elements from set A and set B but elements should not be repeated.
\[ \Rightarrow \begin{array}{*{20}{c}}
  {A \cup B}& = &{\{ 1,2,3,4\} \cup \{ 2,4,6\} }
\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}
  {A \cup B}& = &{\{ 1,2,3,4,6\} }
\end{array}\]
Now according to the given relation, C is a number of sets. All the elements belong to the \[A \cap B\] and \[A \cup B\].
Therefore,
\[ \Rightarrow A \cap B \subseteq C \subseteq A \cup B\]
\[ \Rightarrow \{ 2,4\} \subseteq C \subseteq \{ 1,2,3,4,6\} \]
In the above relation elements 2, 4 are common in \[A \cap B\] and \[A \cup B\]. There are three extra elements that are 1, 3, 6.
therefore, the number of the sets will be \[{2^n}\] . where \[\begin{array}{*{20}{c}}
  n& = &3
\end{array}\]
So, the number of sets = \[{2^3}\]
The number of sets = \[8\].
Therefore, the final answer is 8.
Now, the correct option is (C).

Note: It is important to note that the extra elements will be considered as the value of the number of elements(n). C will be a subset. It means that elements of \[A \cap B\]and \[A \cup B\]belong to the subset C. Hence, we can say that C is a Subset of \[A \cap B\]and \[A \cup B\].