Question

Which of the following is a null set?

• A. \( A = \{ x : x > 1 \text{ and } x < 1 \} \)
• B. \( B = \{ x : x + 3 = 3 \} \)
• C. \( C = \emptyset \)
• D. \( D = \{ x : x \geq 1 \text{ and } x \leq 1 \} \)

Hint:

A null set occurs when no element satisfies the conditions in the set builder notation. Look for contradictory conditions that lead to an empty set.

Answer 1

The Correct Option is A

Step 1: Understand the definition of a null set. 
A null set, or empty set, is a set that contains no elements.
It is denoted by \( \emptyset \) or \( \{\} \). 
Step 2: Analyze the options. 
Option (a): \( A = \{ x : x > 1 \text{ and } x < 1 \} \): This set is empty because there is no value of \( x \) that can satisfy both conditions simultaneously (a number cannot be both greater than 1 and less than 1 at the same time).
Therefore, \( A = \emptyset \), which is a null set.
Option (b): \( B = \{ x : x + 3 = 3 \} \): Solving \( x + 3 = 3 \) gives \( x = 0 \), so this set contains the element 0.
Hence, it is not a null set.
Option (c): \( C = \emptyset \): This is explicitly the null set, as it is empty by definition.
Option (d): \( D = \{ x : x \geq 1 \text{ and } x \leq 1 \} \): This set contains all values of \( x \) that are equal to 1.
Hence, it is not empty; it contains the element 1.
Step 3: Conclusion. 
Option (a) is the null set, as no value of \( x \) can satisfy the condition \( x > 1 \text{ and } x < 1 \) simultaneously.