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!