To solve this problem, we need to understand the definitions of the sets and their complements.
1. Set [tex]\( U \)[/tex]: This is the set of all positive integers.
[tex]\[
U = \{1, 2, 3, 4, \ldots\}
\][/tex]
2. Set [tex]\( A \)[/tex]: This is the subset of [tex]\( U \)[/tex] consisting of all odd positive integers.
[tex]\[
A = \{x \mid x \in U \text{ and } x \text{ is an odd positive integer}\}
\][/tex]
In simpler terms:
[tex]\[
A = \{1, 3, 5, 7, 9, \ldots\}
\][/tex]
3. Complement of Set [tex]\( A \)[/tex] ([tex]\( A^C \)[/tex]): The complement of a set [tex]\( A \)[/tex] with respect to [tex]\( U \)[/tex] is the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
Since [tex]\( A \)[/tex] contains all odd positive integers, [tex]\( A^C \)[/tex] will contain all positive integers that are not in [tex]\( A \)[/tex], which means [tex]\( A^C \)[/tex] will contain all even positive integers.
[tex]\[
A^C = \{x \mid x \in U \text{ and } x \text{ is an even positive integer}\}
\][/tex]
In simpler terms:
[tex]\[
A^C = \{2, 4, 6, 8, 10, \ldots\}
\][/tex]
Given the definition of sets and their complements, we can now see that the correct description that matches [tex]\( A^C \)[/tex] is:
[tex]\[
A^C = \{x \mid x \in U \text{ and is an even positive integer}\}
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{A^C = \{x \mid x \in U \text{ and is an even positive integer}\}} \][/tex]
