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We are asked to find \( A \cap (B \cup C)' \), where \( (B \cup C)' \) is the complement of \( B \cup C \) in the universal set \( U \), and then intersect that set with \( A \).

### Step-by-Step Process:

1. **Find \( B \cup C \):** 
   - From the Venn diagram, \( B \cup C \) includes all elements that are in either \( B \), \( C \), or both.
   - Elements in \( B \) are: \( g, h, x, y \).
   - Elements in \( C \) are: \( x, y, r, z \).
   - Therefore, \( B \cup C = \{ g, h, x, y, r, z \} \).

2. **Find the complement \( (B \cup C)' \):**
   - The complement consists of all elements in \( U \) that are **not** in \( B \cup C \).
   - From the Venn diagram, the universal set \( U \) includes: \( f, g, h, p, q, r, x, y, z \).
   - Removing the elements of \( B \cup C = \{ g, h, x, y, r, z \} \), we are left with \( (B \cup C)' = \{ f, p, q \} \).

3. **Find \( A \cap (B \cup C)' \):**
   - Now, we need to find the intersection of set \( A \) with \( (B \cup C)' \).
   - Elements in \( A \) are: \( f, g, p, x \).
   - Intersecting \( A \) with \( (B \cup C)' = \{ f, p, q \} \), we get \( A \cap (B \cup C)' = \{ f, p \} \).

### Final Answer:
\( A \cap (B \cup C)' = \{ f, p \} \)

Would you like more details on any of these steps or have any questions?

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Here are five related questions you might find useful:
1. What is the difference between the complement and the union of sets?
2. How would the result change if we took the complement of \( A \) instead of \( B \cup C \)?
3. How do you interpret \( A \cap (B \cup C)' \) if the sets were infinite?
4. Can we use De Morgan’s Law to simplify operations with complements in sets?
5. What are common uses of Venn diagrams in probability theory?

**Tip:** Always carefully analyze set complements—sometimes visualizing them in a Venn diagram helps to avoid mistakes.

Find the set A ∩ (B ∪ C)'. Write your answer in roster form or as ∅, given the Venn diagram showing sets A, B, C, and the universal set U.

Learn how to solve a Venn diagram problem involving set intersection and complements: A ∩ (B ∪ C)'. This solution covers the union and complement operations step-by-step and applies De Morgan’s Laws to set theory problems. Suitable for students in Grades 9-10.

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