Question

Which of the following is a null set?
(A) \[A = \{ x:x > 1{\text{ and }}x < 1\} \]
(B) \[B = \{ x:x \geqslant 3\} \]
(C) \[C = \left\{ \phi \right\}\]
(D) \[D = \{ x:x \geqslant 1{\text{ and }}x \leqslant 1\} \]

Answer 1

Hint: The set which does not contain any element is known as the null set. To find which of the given options is a null set we have to check each option one by one and the option in which the resultant set has no elements will be the null set.

Complete step-by-step answer:
To find which of the given options is a null set, we have to check each option one by one and the option in which the resultant set has no elements will be the null set.

We have, first set as \[A = \{ x:x > 1{\text{ and }}x < 1\} \].
It is given that \[x > 1\] and also \[{\text{ }}x < 1\]. But, \[x\] can’t be greater than one and less than one simultaneously. So, there will be no element in \[A\].
Therefore, \[A\] is a null set.

Now, we have a second set as \[B = \{ x:x \geqslant 3\} \].
It is given that \[x \geqslant 3\]. So, there will be an infinite number of elements in this set.
Therefore, \[B\] is not a null set.

Now, we have the third set as \[C = \left\{ \phi \right\}\].
It is given that \[C\] is a set which has no elements. So, there will be one set in which there will be no element.
Therefore, \[C\] is not a null set.

Now, we have the fourth set as \[D = \{ x:x \geqslant 1{\text{ and }}x \leqslant 1\} \].
It is given that \[D\] is a set in which \[x \geqslant 1{\text{ and }}x \leqslant 1\]. So, \[D\] will contain one element i.e., \[1\].
Therefore, \[D\] is not a null set.
So, the correct answer is “Option A and C”.

Note: One important point to note is that null set implies that that there are no elements in a set but here it does not mean that the element is zero i.e., \[\left\{ 0 \right\}\] as if we write \[\left\{ 0 \right\}\] then it is not a null set because this set has an element which is zero. Also, there is only one null set because there is logically only one way that a set can contain nothing.