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=)

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=)

Transcript text: Sets $A, B$, and $C$ are subsets of the universal set $U$. These sets are defined as follows. \[ \begin{array}{l} U=\{f, g, h, p, q, r, x, y, z\} \\ A=\{f, g, p, x\} \\ B=\{g, h, x, y\} \\ C=\{p, r, x, y, z\} \end{array} \] Find $C \cap(B \cup A)^{\prime}$. Write your answer in roster form or as $\varnothing$. \[ C \cap(B \cup A)^{\prime}= \] $\square$

Solution

Solution Steps

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

Compute the union of sets $A$ and $B$.
Find the complement of the union of $A$ and $B$ with respect to the universal set $U$.
Compute the intersection of set $C$ with the complement obtained in step 2.
Write the result in roster form.

Step 1: Compute the Union of Sets $A$ and $B$

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

Step 2: Find the Complement of $A \cup B$ with Respect to $U$

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

Step 3: Compute the Intersection of $C$ with $(A \cup B)^{\prime}$

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