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The given Venn diagram shows three sets: \( A \), \( B \), and \( C \). You are asked to shade the region representing the set:

\[
A \cap (B \cup C)
\]

This expression can be broken down as follows:

1. **\( B \cup C \)**: This represents the union of sets \( B \) and \( C \), meaning all the areas that belong to either \( B \) or \( C \) (or both).

2. **\( A \cap (B \cup C) \)**: This represents the intersection of set \( A \) with the union of \( B \) and \( C \). Therefore, we are looking for the region that belongs to both \( A \) and \( (B \cup C) \).

### Shading Steps:
- First, identify the area corresponding to \( B \cup C \). This includes all regions inside either the \( B \) or \( C \) circles.
- Next, find the intersection of this region with set \( A \). This is the part of the union that also lies within the \( A \) circle.

In terms of the Venn diagram:
- Shade the regions where sets \( A \) and either \( B \) or \( C \) overlap.

Would you like a more detailed breakdown or have any questions about the logic?

Here are five questions to expand this concept:
1. How would the result change if we had \( A \cup (B \cap C) \) instead?
2. What is the region corresponding to \( (A \cap B) \cup C \)?
3. Can you explain the difference between set intersection and set union?
4. How would this diagram look if you shaded \( A \cap B \cap C \)?
5. What is the result of \( A \cup (B \cup C) \) in terms of the diagram?

**Tip:** When working with set notation, breaking down complex expressions into smaller components (like unions and intersections) helps clarify what areas need to be shaded.

Learn how to shade the Venn diagram representing A ∩ (B ∪ C), where set A intersects with the union of sets B and C. Understand the step-by-step breakdown of this set theory problem, suitable for students in Grades 9-12.

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