Question

Which of the following is a null set?

• A. $X=\left\{\phi \right\}$
• B. $Y=\left\{X:X\ge 3\right\}$
• C. $Z=\left\{X:X<1\mathrm{and}X>1\right\}$
• D. $A=\left\{X:X\ge 1\mathrm{and}X\le 1\right\}$

Answer 1

The correct option is C

$Z=\left\{X:X<1\mathrm{and}X>1\right\}$

Explanation for correct option:

Determine the null set:

The null set is an empty set.

The set $Z$ can not have any element as no number is less than 1 and greater than 1 at the same time.

So, $Z$ is an empty set.

Consequently, option C is correct.

Explanation for incorrect options:

Option A: $X=\left\{\phi \right\}$

The set $X$ has one element that is $\phi$, so it is not empty.

So, option A is incorrect.

Option B: $Y=\left\{X:X\ge 3\right\}$

The set $Y$ can have multiple elements greater than 3.

Some of the elements in $Y$ are $4,6,7...$ So, it is also not empty.

So, option B is incorrect.

Option D: $A=\left\{X:X\ge 1\mathrm{and}X\le 1\right\}$

The set $A$ has one elements that is 1. So, $A$ is also a non-empty set.

So, option D is also incorrect.

Thus, the correct option is C