Question

Which of the following is an empty set ?

• A. {x : x² - 9 = 0, x ∈ R}
• B. {x : x² - 1 = 0, x ∈ R}
• C. {x : x² = x + 2, x ∈ R}
• D. {x : x² + 1 = 0, x ∈ R}

Hint:

When solving equations to identify empty sets, remember that if the equation leads to a condition that has no solution in the given set (in this case, real numbers), the set is empty.

Answer 1

The Correct Option is D

The correct answer is: (D): \( \{ x : x^2 + 1 = 0, x \in \mathbb{R} \} \)

We are tasked with identifying which of the following sets is an empty set. Let’s analyze each option:

Step 1: Examine the set \( \{ x : x^2 + 1 = 0, x \in \mathbb{R} \} \)

This set contains all real numbers \( x \) that satisfy the equation \( x^2 + 1 = 0 \). Let’s solve this equation:

\[ x^2 + 1 = 0 \quad \Rightarrow \quad x^2 = -1 \]

There is no real number \( x \) whose square is negative. Therefore, there are no solutions to the equation \( x^2 + 1 = 0 \) in the set of real numbers \( \mathbb{R} \). This implies that the set is empty.

Step 2: Conclusion
Since no real number satisfies the given equation, the set \( \{ x : x^2 + 1 = 0, x \in \mathbb{R} \} \) is indeed an empty set.

Conclusion:
The correct answer is (D): \( \{ x : x^2 + 1 = 0, x \in \mathbb{R} \} \), as it is the empty set.