To determine the set [tex]\((A \cap C)^{\prime}\)[/tex], we follow these steps:
1. Identify the universal set [tex]\(U\)[/tex] and the sets [tex]\(A\)[/tex] and [tex]\(C\)[/tex] given in the problem.
[tex]\[
U = \{1, 2, 3, 4, 5, 6, 7, 8\}
\][/tex]
[tex]\[
A = \{2, 4, 7, 8\}
\][/tex]
[tex]\[
C = \{3, 4, 5, 7, 8\}
\][/tex]
2. Calculate the intersection of sets [tex]\(A\)[/tex] and [tex]\(C\)[/tex] ([tex]\(A \cap C\)[/tex]).
[tex]\[
A \cap C = \{4, 7, 8\}
\][/tex]
3. Find the complement of the intersection set [tex]\((A \cap C)\)[/tex] with respect to the universal set [tex]\(U\)[/tex]. The complement [tex]\( (A \cap C)^{\prime} \)[/tex] is the set of elements that are in [tex]\(U\)[/tex] but not in [tex]\(A \cap C\)[/tex].
[tex]\[
(A \cap C)^{\prime} = U - (A \cap C)
\][/tex]
Elements in [tex]\(U\)[/tex] that are not in [tex]\(A \cap C\)[/tex]:
[tex]\[
(A \cap C)^{\prime} = \{1, 2, 3, 5, 6\}
\][/tex]
4. Thus, the set [tex]\( (A \cap C)^{\prime} \)[/tex] is:
[tex]\[
(A \cap C)^{\prime} = \{1, 2, 3, 5, 6\}
\][/tex]
Select the correct choice:
A. [tex]\((A \cap C)^{\prime} = \{1, 2, 3, 5, 6\}\)[/tex]
