Question

algebra i b-cr
pre-test complete
1 2
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 universal set

$U = \{x|x < 0, x \in \mathbb{R}\}$ (all negative real numbers)

Step2: Evaluate Option 1

Negative numbers have negative cube roots (e.g., $\sqrt[3]{-8}=-2$). This set equals $U$, not empty.

Step3: Evaluate Option 2

Square roots of negative real numbers are not real numbers. Since $U$ only contains real numbers, no $x \in U$ satisfies this. This set is empty.

Step4: Evaluate Option 3

Product of positive number and $-1$ is negative. This set equals $U$, not empty.

Step5: Evaluate Option 4

Sum of negative and positive can be negative (e.g., $-5+2=-3$). This set is non-empty.

Answer:

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