Question

Find the number of proper subsets of A, $\text{A}=\left\{ 1,2,\left\{ 3,4 \right\} \right\}$
$\begin{align}
  & \left( A \right)\text{ 16} \\
 & \left( B \right)\text{ 15} \\
 & \left( C \right)\text{ 3} \\
 & \left( D \right)\text{ 6} \\
\end{align}$

Answer 1

Hint: We solve the problem by finding the possible subsets of given set. Then we use the definition of a proper subset and we find all the proper subsets that can be formed from the given set in the question. So, we can find the number of proper subsets of the given set.

Complete step by step answer:
First, let us recall the definition of a proper subset.
For any set A, proper subset of A is a subset that is not an empty set or itself.
We are given that $\text{A}=\left\{ 1,2,\left\{ 3,4 \right\} \right\}$.
Then we start by finding all the subsets of A.
So, the possible subsets of A are,
As empty set is a subset of every set, $\left\{ {} \right\}$ is a subset of A.
Taking one element of A at a time the subsets we can form are
$\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3,4 \right\}$.
Taking two elements of A at a time the subsets we can form are
$\left\{ 1,2 \right\},\left\{ 1,\left\{ 3,4 \right\} \right\},\left\{ 2,\left\{ 3,4 \right\} \right\}$.

Taking three elements of A at a time the subsets we can form are $\left\{ 1,2,\left\{ 3,4 \right\} \right\}$ which is A itself.
So, all the possible subsets we get from A are
$\left\{ {} \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3,4 \right\},\left\{ 1,2 \right\},\left\{ 1,\left\{ 3,4 \right\} \right\},\left\{ 2,\left\{ 3,4 \right\} \right\},\left\{ 1,2,\left\{ 3,4 \right\} \right\}$ that is a total of 8 subsets can be formed from the given set A.
But, from the definition of the proper subset that we have discussed above, we need to subtract the empty subset and the subset A itself from the total number of subsets.
So, Total number of proper subsets that can be formed is equal to
$\begin{align}
  & \Rightarrow 8-2 \\
 & \Rightarrow 6 \\
\end{align}$
Hence, number of proper sets that can be formed from set A is equal to 6.

So, the correct answer is “Option D”.

Note: The major mistake that one makes while solving this question is taking $\left\{ 3,4 \right\}$ as two elements $\left\{ 3 \right\},\left\{ 4 \right\}$. But as they are given as one set, we should not separate them into different elements. So, we need to consider $\left\{ 3,4 \right\}$ as a single element.