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 the first option

For a negative real number \( x \in U \), the cube root of a negative number is negative (since \( \sqrt[3]{-a}=-\sqrt[3]{a} \) for \( a>0 \)). So there are elements in this set.

Step2: Analyze the second option

The square root of a real number is defined as a non - negative number (in the set of real numbers, \( \sqrt{x} \geq0 \) when \( x\geq0 \), and for \( x < 0 \), the square root is not a real number). So if \( x\in U \) (i.e., \( x<0 \)), there is no real number \( y \) such that \( y^{2}=x \) and \( y < 0 \) (because the square of a real number is non - negative). So this set has no elements.

Step3: Analyze the third option

If \( x \) is the product of a positive number \( a(a > 0) \) and \( - 1 \), then \( x=-a \), which is a negative real number. So \( x\in U \), and this set is non - empty.

Step4: Analyze the fourth option

Let the negative number be \( -m(m>0) \) and the positive number be \( n(n > 0) \). Then their sum is \( n - m \). If \( n

Answer:

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