Question

Which of the following does not have a proper subset?
\[\begin{align}
  & A.\left\{ x:x\in Q \right\} \\
 & B.\left\{ x:x\in N,3\text{ }<\text{ }x\text{ }<\text{ }4 \right\} \\
 & C.\left\{ x:x\in Q,3\text{ }<\text{ }x\text{ }<\text{ }4 \right\} \\
 & D.N\text{one of these} \\
\end{align}\]

Answer 1

Hint: In this question, we are given three sets and we need to find the set which does not have any proper subset. Proper subsets are the subset of the set which are neither equal to the given set nor an empty set. For solving, we will try to understand the sets and then try to find their proper subsets.

Complete step-by-step solution
Here we are given three sets.
As we know, proper subsets are the subsets of the given set which are neither equal to the given set nor an empty set. So let us first understand the given set clearly and then try to find their proper subsets.
For set A $\left\{ x:x\in Q \right\}$.
It is a set of all rational numbers and hence it cannot be written in roster form. But the set of rational numbers has a proper subset. For example: $\left\{ \dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5} \right\}$ is a subset of the set of rational numbers which is not empty and does not cover all rational numbers. So option A is not the correct answer.
For set B $\left\{ x:x\in N,3\text{ }<\text{ }x\text{ }<\text{ }4 \right\}$.
It is a set consisting of all-natural numbers lying between 3 and 4. Since there are no natural numbers between 3 and 4 so this set is an empty set. Thus it cannot have any proper subset. So option B is the correct answer.
Let us check set C also set C is $\left\{ x:x\in Q,3\text{ }<\text{ }x\text{ }<\text{ }4 \right\}$.
It is a set consisting of all rational numbers between 3 and 4. Since there are infinite rational numbers between 3 and 4, so it can have any proper subset. For example: $\left\{ \dfrac{13}{4},\dfrac{7}{2},\dfrac{15}{4} \right\}$ is a proper subset of C. So option C is not the correct answer.
Hence, option B is the correct answer.

Note: Students should note that here Q means the set of rational numbers, N means the set of natural numbers. These sets cannot be written in roster form as we cannot write all the rational numbers. All sets have subsets but not every set has a proper subset. If there was a sign of equality for set B then we have a proper subset.