Let's analyze each set to determine which one is an empty set.
1. [tex]\(\left\{ x \mid x \in U \text{ and } \frac{1}{2} x \text{ is prime} \right\}\)[/tex]:
For this set, we need to check if [tex]\(\frac{1}{2} x\)[/tex] is a prime number. This means [tex]\(x\)[/tex] must be an even number since [tex]\(\frac{1}{2} x\)[/tex] must be an integer. Let's consider some even [tex]\(x\)[/tex]:
- If [tex]\(x = 4\)[/tex], then [tex]\(\frac{1}{2} \times 4 = 2\)[/tex], which is prime.
- If [tex]\(x = 6\)[/tex], then [tex]\(\frac{1}{2} \times 6 = 3\)[/tex], which is prime.
Therefore, this set is not empty.
2. [tex]\(\{x \mid x \in U \text{ and } 2x \text{ is prime}\}\)[/tex]:
For this set, we need to check if [tex]\(2x\)[/tex] is a prime number. For [tex]\(2x\)[/tex] to be prime, [tex]\(x\)[/tex] itself must be a fraction since any positive integer [tex]\(x\)[/tex] times 2 is greater than 2 and even (except if [tex]\(x = 1\)[/tex], but [tex]\(1\)[/tex] is not in our set [tex]\(U\)[/tex]). Thus, there are no positive integers greater than 1 that satisfy this condition.
Therefore, this set is empty.
3. [tex]\(\left\{ x \mid x \in U \text{ and } \frac{1}{2} x \text{ can be written as a fraction} \right\}\)[/tex]:
For this set, we need to check if [tex]\(\frac{1}{2} x\)[/tex] can be written as a fraction. Since [tex]\(\frac{1}{2} x\)[/tex] simplifies to any value divided by 2 (and all [tex]\(x \in U\)[/tex] are integers), it is always a fraction.
Therefore, this set is not empty.
4. [tex]\(\{x \mid x \in U \text{ and } 2x \text{ can be written as a fraction}\}\)[/tex]:
For this set, we need to check if [tex]\(2x\)[/tex] can be written as a fraction. Since [tex]\(x \in U\)[/tex] are integers and multiplying any integer by 2 yields another integer, it can be written as a fraction (ex: [tex]\(\frac{2x}{1}\)[/tex]).
Therefore, this set is not empty.
By analyzing each set, we find that the empty set is:
[tex]\[
\{x \mid x \in U \text{ and } 2x \text{ is prime}\}
\][/tex]
