Question

The set $\left( x:x\ne x \right)$ may be equal to
A. $\left\{ 0 \right\}$
B. $\left\{ 1 \right\}$
C. $\left\{ 3 \right\}$
D. $\left\{ {} \right\}$

Answer 1

Hint: We first try to define the difference between the null set and $\left\{ 0 \right\}$. Then we try to figure out the elements of the set $\left( x:x\ne x \right)$. We place the set in its respective option based on the number of elements it has and solves the problem.

Complete step-by-step solution:
First, we need to define the difference between two sets $\left\{ 0 \right\}$ and $\left\{ {} \right\}$.
In the case of $\left\{ {} \right\}$, it means the null set. There is no element in the set and if $A=\left\{ {} \right\}$ then $n\left( A \right)=0$. We also define this null set as $\phi $.
On the other hand we have $\left\{ 0 \right\}$ which means this set has an element which is 0 and if $B=\left\{ 0 \right\}$ then $n\left( B \right)=1$.
Now we check the given set $\left( x:x\ne x \right)$. It is irrelevant to the domain of x. Whatever be the value of x it will always be equal to itself. So, there exists no such x for which the set $\left( x:x\ne x \right)$ exists.
Number of elements in that set is 0. So, the set is null. The correct option is D.

Note: Having a second bracket always defines a set. But if we get such brackets inside another bracket then the first one defines the elements of the second one. For example: if we take $A=\left\{ 2,\left\{ 5,6 \right\},8 \right\}$. It means 5 and 6 are not elements of A rather $\left\{ 5,6 \right\}$ is an element of A.