Let's analyze each set description and determine whether it is empty or not.
1. [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative cube root}\}\)[/tex]:
- [tex]\(U\)[/tex] is the set of negative real numbers.
- A negative number [tex]\(x\)[/tex] can have a negative cube root because the cube root of a negative number is also negative (e.g., [tex]\(\sqrt[3]{-8} = -2\)[/tex]).
- Therefore, this set contains all negative real numbers because every negative real number in [tex]\(U\)[/tex] has a negative cube root. This set is not empty.
2. [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative square root}\}\)[/tex]:
- [tex]\(U\)[/tex] is the set of negative real numbers.
- The square root of a negative number is not a real number; it is an imaginary number (e.g., [tex]\(\sqrt{-4} = 2i\)[/tex]).
- Since [tex]\(x\)[/tex] must be a real number and we are considering real numbers only, there are no real numbers in [tex]\(U\)[/tex] that have a real square root, let alone a negative square root.
- Therefore, this set is empty.
3. [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the product of a positive number and -1}\}\)[/tex]:
- [tex]\(U\)[/tex] is the set of negative real numbers.
- A number [tex]\(x\)[/tex] that is the product of a positive number and [tex]\(-1\)[/tex] is negative (e.g., [tex]\(x = -1 \cdot 3 = -3\)[/tex]).
- All numbers in [tex]\(U\)[/tex] are negative by definition, and they can be expressed as the product of a positive number and [tex]\(-1\)[/tex].
- Therefore, this set contains all negative real numbers in [tex]\(U\)[/tex] and is not empty.
4. [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the sum of one negative and one positive number}\}\)[/tex]:
- [tex]\(U\)[/tex] is the set of negative real numbers.
- A negative number can be expressed as the sum of a negative number and a (larger in magnitude) negative number or a smaller positive number (e.g., [tex]\(-5 = -8 + 3\)[/tex]).
- All numbers in [tex]\(U\)[/tex] can be expressed as the sum of a smaller positive number and a larger negative number.
- Therefore, this set is not empty.
So, the only set that is empty is:
[tex]\[
\{x \mid x \in U \text{ and } x \text{ has a negative square root}\}
\][/tex]
