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The problem involves sets \( A \), \( B \), and \( C \), and we need to find the union of \( A \) with the intersection of \( B \) and \( C \). Here's how to approach the solution step by step:

### Given sets:
- \( A = \{1, 2, 4, 5, 6, 9, 10, 11, 14\} \)
- \( B = \{3, 4, 7, 8, 9, 11\} \)
- \( C = \{1, 5, 10\} \)

### Universal set:
\( U = \{1, 2, 3, 4, 5, \dots, 15\} \)

### Step 1: Find \( B \cap C \) (Intersection of \( B \) and \( C \))
The intersection \( B \cap C \) consists of elements that are common to both \( B \) and \( C \). We compare the elements:

- \( B = \{3, 4, 7, 8, 9, 11\} \)
- \( C = \{1, 5, 10\} \)

There are **no common elements** between \( B \) and \( C \). Hence,
\[
B \cap C = \emptyset
\]

### Step 2: Find \( A \cup (B \cap C) \)
Since \( B \cap C = \emptyset \), the union of \( A \) with \( B \cap C \) will simply be \( A \), as the union of a set with the empty set is the set itself.

Thus,
\[
A \cup (B \cap C) = A = \{1, 2, 4, 5, 6, 9, 10, 11, 14\}
\]

### Conclusion:
The answer is \( \{1, 2, 4, 5, 6, 9, 10, 11, 14\} \), which matches the third option.

---

Would you like further clarification on any part of this process? 

Here are five related questions to consider:
1. How do you calculate the intersection of two sets?
2. What is the result of the union of a set with the empty set?
3. How does the universal set play a role in set operations?
4. Can sets contain repeated elements, or is each element unique?
5. How would the result change if \( B \cap C \) were non-empty?

**Tip:** Always look for common elements when calculating intersections—this helps ensure accuracy in set operations!

If A = {1, 2, 4, 5, 6, 9, 10, 11, 14}, B = {3, 4, 7, 8, 9, 11} and C = {1, 5, 10} have universal set U = {1, 2, 3, 4,..., 15}. Then A ∪ (B ∩ C) = ?

Solve a set theory problem involving sets A, B, and C with a universal set U. Learn how to find the intersection and union of sets, applying key concepts like union and intersection. Suitable for high school students.

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