To determine which sets are empty, we must analyze each given condition in the context of the set [tex]\( U \)[/tex] of negative real numbers. Here is a step-by-step analysis of each condition:
1. Set 1: [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative cube root}\}\)[/tex]
- The cube root function can be negative for negative real numbers. For example, the cube root of [tex]\(-8\)[/tex] is [tex]\(-2\)[/tex]. This means there exist negative real numbers whose cube root is also negative.
- Therefore, this set is not empty.
2. Set 2: [tex]\(\{x \mid x \in U \text{ and } x \text{ has a negative square root}\}\)[/tex]
- For any real number, the square root is defined to be non-negative. The square root of a negative number is not real, it is imaginary.
- Since [tex]\( x \in U \)[/tex] consists only of negative real numbers and no negative real number has a real (negative or positive) square root, this set is empty.
3. Set 3: [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the product of a positive number and -1}\}\)[/tex]
- Any negative real number [tex]\(x\)[/tex] can be written as [tex]\(-a\)[/tex] where [tex]\(a\)[/tex] is a positive real number.
- Therefore, any [tex]\( x \in U \)[/tex] can be expressed as [tex]\(-a\)[/tex], where [tex]\( a > 0 \)[/tex].
- This means that this set is not empty.
4. Set 4: [tex]\(\{x \mid x \in U \text{ and } x \text{ is equal to the sum of one negative and one positive number}\}\)[/tex]
- For any [tex]\( x \in U \)[/tex], we can always find a negative number [tex]\(y < 0\)[/tex] and a positive number [tex]\(z > 0\)[/tex] such that [tex]\( x = y + z \)[/tex]. For example, for [tex]\( x = -5 \)[/tex], we can have [tex]\( y = -7 \)[/tex] and [tex]\( z = 2 \)[/tex] such that [tex]\( x = -7 + 2 = -5 \)[/tex].
- Thus, this set is not empty.
In conclusion, the only empty set is:
[tex]\[ \{x \mid x \in U \text{ and } x \text{ has a negative square root}\}. \][/tex]
