Question

consider u = {x|x is a negative real number}. which is an empty set? {x|x ∈ u and x has a negative cube root} {x|x ∈ u and x has a negative square root} {x|x ∈ u and x is equal to the product of a positive number and −1} {x|x ∈ u and x is equal to the sum of one negative and one positive number}

Explanation:

Step1: Analyze each option

• For the first option: Let \( x \in U \), so \( x < 0 \). The cube root of a negative number is negative (e.g., \( \sqrt[3]{-8}=-2 \)). So there are elements in this set.
• For the second option: Let \( x \in U \), so \( x < 0 \). The square root of a negative number is not a real number (in the set of real numbers, \( \sqrt{x} \) is defined only for \( x\geq0 \)). So there are no real numbers \( x \) in \( U \) such that \( x \) has a negative square root (since square root of negative real number is not real).
• For the third option: If \( x \) is the product of a positive number and - 1, then \( x=-a \) where \( a > 0 \), so \( x < 0 \), which means \( x\in U \). So this set has elements.
• For the fourth option: The sum of a negative and a positive number can be negative (e.g., \( -5 + 3=-2 \in U \)). So this set has elements.

Answer:

\(\{x|x\in U\text{ and }x\text{ has a negative square root}\}\)