https://file.aiquickdraw.com/mathbot/file/17252548849227dx5i9fb.jpg

(A intersection B) intersection (A union C)

Let's analyze the provided Venn diagram to find \((A \cap B) \cap (A \cup C)\).

### Step 1: Breakdown of the Sets
- **Region 5** is where sets \(A\) and \(B\) intersect.
- The union \(A \cup C\) includes all elements in both \(A\) and \(C\), which corresponds to regions 1, 2, 4, 5, and 7.

### Step 2: Identify \(A \cap B\)
- The intersection of sets \(A\) and \(B\) corresponds to region **5**.

### Step 3: Identify \(A \cup C\)
- The union of sets \(A\) and \(C\) covers regions **1, 2, 4, 5, and 7**.

### Step 4: Find \((A \cap B) \cap (A \cup C)\)
- The common region between \(A \cap B\) (which is region 5) and \(A \cup C\) (which includes regions 1, 2, 4, 5, and 7) is only **region 5**.

### Conclusion:
The result of \((A \cap B) \cap (A \cup C)\) is the region labeled **5**.

Would you like more details or clarification? Here are some related questions:

1. What would \((A \cap B) \cup (A \cap C)\) represent in this diagram?
2. How would you represent the complement of \((A \cup B) \cap C\)?
3. How does the universal set \(U\) impact set operations in this diagram?
4. What is the result of \((A \cap B^c) \cup C\)?
5. How would the diagram change if the sets were disjoint?

**Tip:** When working with complex Venn diagrams, try breaking down the problem into simpler set operations to make it easier to analyze.

Explore the set theory problem (A intersection B) intersection (A union C) with a detailed analysis. Learn how to interpret complex set intersections and unions, and apply them to solve problems effectively. Suitable for college-level mathematics.

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