To solve for $\left( A \cap C \cap B' \right) \cup B$, we'll go through the problem step by step.

Given Sets:

• Universal set $U = \{f, g, h, p, q, r, x, y, z\}$
• Set $A = \{f, g, h, q, r\}$
• Set $B = \{q, r, x, z\}$
• Set $C = \{h, p, q, z\}$

Step 1: Find the complement of $B$, denoted as $B'$

The complement of $B$ contains all elements in the universal set $U$ that are not in $B$:

$B' = U \setminus B = \{f, g, h, p, y\}$

Step 2: Find the intersection of $A \cap C$

We now find the common elements between sets $A$ and $C$:

$A = \{f, g, h, q, r\}$ $C = \{h, p, q, z\}$

The intersection $A \cap C$ is:

$A \cap C = \{h, q\}$

Step 3: Find the intersection of $A \cap C \cap B'$

Now we intersect $A \cap C = \{h, q\}$ with $B' = \{f, g, h, p, y\}$:

$A \cap C \cap B' = \{h\}$

Step 4: Find the union of $(A \cap C \cap B') \cup B$

Now we take the union of $\{h\}$ and $B = \{q, r, x, z\}$:

$(A \cap C \cap B') \cup B = \{h\} \cup \{q, r, x, z\} = \{h, q, r, x, z\}$

Final Answer:

$\left( A \cap C \cap B' \right) \cup B = \{h, q, r, x, z\}$

This is the result in roster form.

Would you like further clarification or details on any of the steps?

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Here are 5 related questions to deepen understanding:

1. What is the result of $A \cup B$?
2. How do you compute $A \cap B'$?
3. What is the complement of set $C$ in the universal set $U$?
4. Can you find the intersection $B \cap C$?
5. What is $(A \cup B) \cap (A \cup C)$?

Tip: When working with set operations, drawing a Venn diagram can help visualize intersections and unions more clearly.