To solve the problems, we need to manipulate the given sets and apply set operations. Let's break down the solutions step-by-step.
### Problem (a): [tex]\((A \cup B)^{\prime}\)[/tex]
1. Find [tex]\( A \cup B \)[/tex]:
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] consists of all elements that are in [tex]\( A \)[/tex], in [tex]\( B \)[/tex], or in both.
[tex]\[
A \cup B = \{ f, q, x, z \} \cup \{ f, m, q \} = \{ f, q, x, z, m \}
\][/tex]
So, [tex]\( A \cup B = \{ f, q, x, z, m \} \)[/tex].
2. Find [tex]\((A \cup B)^{\prime}\)[/tex]:
The complement of [tex]\( A \cup B \)[/tex] consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \cup B \)[/tex].
[tex]\[
(A \cup B)^{\prime} = U - (A \cup B) = \{ f, k, m, q, x, z \} - \{ f, q, x, z, m \} = \{ k \}
\][/tex]
So, [tex]\((A \cup B)^{\prime} = \{ k \}\)[/tex].
### Problem (b): [tex]\(A^{\prime} \cap B\)[/tex]
1. Find [tex]\( A^{\prime} \)[/tex]:
The complement of [tex]\( A \)[/tex] consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
[tex]\[
A^{\prime} = U - A = \{ f, k, m, q, x, z \} - \{ f, q, x, z \} = \{ k, m \}
\][/tex]
So, [tex]\( A^{\prime} = \{ k, m \} \)[/tex].
2. Find [tex]\( A^{\prime} \cap B \)[/tex]:
The intersection of [tex]\( A^{\prime} \)[/tex] and [tex]\( B \)[/tex] consists of all elements that are in both [tex]\( A^{\prime} \)[/tex] and [tex]\( B \)[/tex].
[tex]\[
A^{\prime} \cap B = \{ k, m \} \cap \{ f, m, q \} = \{ m \}
\][/tex]
So, [tex]\( A^{\prime} \cap B = \{ m \} \)[/tex].
### Final Answers
(a) [tex]\((A \cup B)^{\prime} = \{ k \}\)[/tex]
(b) [tex]\(A^{\prime} \cap B = \{ m \}\)[/tex]
