The question is based on set theory, involving unions and intersections of sets. Let's break it down step by step:

Given:

• Universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
• Set $A = \{2, 3, 4, 5, 9\}$
• Set $B = \{3, 7, 8\}$
• Set $C = \{1, 2, 5, 6\}$

1. First, we need to find $A \cup (B \cap C)$:

Step 1: Find $B \cap C$ (intersection of sets $B$ and $C$):

• $B = \{3, 7, 8\}$
• $C = \{1, 2, 5, 6\}$
• There are no common elements between $B$ and $C$, so: $B \cap C = \emptyset \ (\text{empty set})$

Step 2: Find $A \cup (B \cap C)$ (union of $A$ and $\emptyset$):

• Since the union of any set with an empty set is the set itself, we have: $A \cup (B \cap C) = A = \{2, 3, 4, 5, 9\}$
• So, $A \cup (B \cap C) = \{2, 3, 4, 5, 9\}$

2. Next, find $(A \cup B) \cap (A \cup C)$:

Step 1: Find $A \cup B$ (union of sets $A$ and $B$):

• $A = \{2, 3, 4, 5, 9\}$
• $B = \{3, 7, 8\}$
• $A \cup B = \{2, 3, 4, 5, 7, 8, 9\}$

Step 2: Find $A \cup C$ (union of sets $A$ and $C$):

• $A = \{2, 3, 4, 5, 9\}$
• $C = \{1, 2, 5, 6\}$
• $A \cup C = \{1, 2, 3, 4, 5, 6, 9\}$

Step 3: Find $(A \cup B) \cap (A \cup C)$ (intersection of $A \cup B$ and $A \cup C$):

• $A \cup B = \{2, 3, 4, 5, 7, 8, 9\}$
• $A \cup C = \{1, 2, 3, 4, 5, 6, 9\}$
• Common elements are $\{2, 3, 4, 5, 9\}$, so: $(A \cup B) \cap (A \cup C) = \{2, 3, 4, 5, 9\}$

Final Answers:

• $A \cup (B \cap C) = \{2, 3, 4, 5, 9\}$
• $(A \cup B) \cap (A \cup C) = \{2, 3, 4, 5, 9\}$

Would you like further details or have any questions? Here are five questions to expand on this topic:

1. What is the difference between union and intersection in set theory?
2. How do we find the complement of a set within a universal set?
3. Can you explain what a Venn diagram looks like for three sets?
4. How does the empty set impact operations like union and intersection?
5. What are the properties of set theory such as distributive and associative properties?

Tip: Drawing Venn diagrams can help visualize set operations and simplify complex problems.