Let's analyze the problem step-by-step.
1. Define the Universal Set [tex]\( U \)[/tex]:
- Universal set [tex]\( U \)[/tex] contains all positive integers. These are the numbers \{1, 2, 3, 4, 5, \ldots\}.
2. Define Set [tex]\( A \)[/tex]:
- Set [tex]\( A \)[/tex] is described as [tex]\(\{x \mid x \in U\)[/tex] and [tex]\(x\)[/tex] is an odd positive integer\}\).
- Therefore, [tex]\( A \)[/tex] includes all positive odd integers: \{1, 3, 5, 7, 9, \ldots\}.
3. Finding the Complement of Set [tex]\( A \)[/tex] in [tex]\( U \)[/tex]:
- The complement [tex]\( A^c \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
- Since [tex]\( A \)[/tex] contains all odd positive integers, [tex]\( A^c \)[/tex] should contain all the positive integers that are not odd.
- Therefore, [tex]\( A^c \)[/tex] must include all positive integers that are even.
4. Describe [tex]\( A^c \)[/tex]:
- Based on the previous step, [tex]\( A^c \)[/tex] is the set of all positive even integers: \{2, 4, 6, 8, 10, \ldots\}.
5. Choose the Correct Option:
- Verify each given option:
- Option 1: [tex]\( A^c = \{x \mid x \in U \text{ and is a negative integer}\} \)[/tex] (Incorrect: [tex]\( U \)[/tex] only contains positive integers).
- Option 2: [tex]\( A^c = \{x \mid x \in U \text{ and is zero}\} \)[/tex] (Incorrect: Zero is not a positive integer, hence not in [tex]\( U \)[/tex]).
- Option 3: [tex]\( A^c = \{x \mid x \in U \text{ and is not an integer}\} \)[/tex] (Incorrect: [tex]\( U \)[/tex] only contains integers).
- Option 4: [tex]\( A^c = \{x \mid x \text{ is an even positive integer}\} \)[/tex] (Correct: Matches the description we derived).
Therefore, the correct description of the complement of set [tex]\( A \)[/tex], denoted as [tex]\( A^c \)[/tex], is:
[tex]\[ A^c = \{x \mid x \in U \text{ and is an even positive integer}\} \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{4} \][/tex]
