Questions: Sets (A) and (B) are subsets of the universal set (U). These sets are defined as follows. [ U=f, m, q, r, y, z ] [ A=f, m, r, y ] [ B=m, q, y ] Find the following sets. Write your answer in roster form or as (varnothing). (a) ((A cup B)^prime=) (square) (b) (A^prime cap B=) (square)

Find \((A \cup B)^{\prime}\).

Step 1: Find \(A \cup B\).

\(A \cup B\) is the union of sets \(A\) and \(B\).
Given:
\(A = \{f, m, r, y\}\)
\(B = \{m, q, y\}\)
Thus, \(A \cup B = \{f, m, q, r, y\}\).

Step 2: Find the complement of \(A \cup B\), denoted as \((A \cup B)^{\prime}\).

The complement of a set \(S\) with respect to the universal set \(U\) is \(U \setminus S\).
Given:
\(U = \{f, m, q, r, y, z\}\)
\(A \cup B = \{f, m, q, r, y\}\)
Thus, \((A \cup B)^{\prime} = U \setminus (A \cup B) = \{z\}\).

\(\boxed{(A \cup B)^{\prime} = \{z\}}\)

Find \(A^{\prime} \cap B\).

Step 1: Find the complement of \(A\), denoted as \(A^{\prime}\).

The complement of \(A\) with respect to the universal set \(U\) is \(U \setminus A\).
Given:
\(U = \{f, m, q, r, y, z\}\)
\(A = \{f, m, r, y\}\)
Thus, \(A^{\prime} = U \setminus A = \{q, z\}\).

Step 2: Find the intersection of \(A^{\prime}\) and \(B\).

Given:
\(A^{\prime} = \{q, z\}\)
\(B = \{m, q, y\}\)
Thus, \(A^{\prime} \cap B = \{q\}\).

\(\boxed{A^{\prime} \cap B = \{q\}}\)

\(\boxed{(A \cup B)^{\prime} = \{z\}}\)
\(\boxed{A^{\prime} \cap B = \{q\}}\)