Questions: Sets (A, B), and (C) are subsets of the universal set (U). These sets are defined as follows. (U=f, g, h, p, q, r, x, y, z) (A=f, g, p, x) (B=g, h, x, y) (C=p, r, x, y, z) Find (C cap(B cup A)^prime). Write your answer in roster form or as (varnothing). (C cap(B cup A)^prime=)

Solution Steps

To solve the problem, we need to follow these steps:

1. Compute the union of sets \(A\) and \(B\).
2. Find the complement of the union of \(A\) and \(B\) with respect to the universal set \(U\).
3. Compute the intersection of set \(C\) with the complement obtained in step 2.
4. Write the result in roster form.

First, we find the union of sets \(A\) and \(B\): \[ A \cup B = \{f, g, p, x\} \cup \{g, h, x, y\} = \{f, g, p, x, y, h\} \]

Next, we find the complement of \(A \cup B\) with respect to the universal set \(U\): \[ (A \cup B)^{\prime} = U \setminus (A \cup B) = \{f, g, h, p, q, r, x, y, z\} \setminus \{f, g, p, x, y, h\} = \{q, z, r\} \]

Finally, we compute the intersection of set \(C\) with the complement of \(A \cup B\): \[ C \cap (A \cup B)^{\prime} = \{p, r, x, y, z\} \cap \{q, z, r\} = \{z, r\} \]

Final Answer