To solve $(A \cap B) \cup C$ using the given sets, let's follow these steps:

Given sets:

• $A = \{4, 5, 6, 7, 8, 9\}$
• $B = \{1, 3, 5, 7, 9, 11, 13, 15\}$
• $C = \{2, 5, 6, 9, 10, 12\}$

Step 1: Find $A \cap B$ (Intersection of A and B)

The intersection of two sets is the set of elements common to both.

• Common elements between $A$ and $B$ are: $\{5, 7, 9\}$

Thus, $A \cap B = \{5, 7, 9\}$.

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

The union of two sets is the set of all distinct elements that are in either of the sets.

• Elements of $(A \cap B) = \{5, 7, 9\}$
• Elements of $C = \{2, 5, 6, 9, 10, 12\}$

The union of $\{5, 7, 9\}$ and $\{2, 5, 6, 9, 10, 12\}$ is:

• $\{2, 5, 6, 7, 9, 10, 12\}$

Final Answer:

$(A \cap B) \cup C = \{2, 5, 6, 7, 9, 10, 12\}$

Let me know if you would like further details or have any questions!

Follow-up Questions:

1. How do you find the difference between two sets, such as $A - B$?
2. What is the symmetric difference of sets, and how does it differ from the union?
3. How can you represent $A \cap B$ using a Venn diagram?
4. What are the properties of set operations like commutativity or associativity in intersections and unions?
5. Can you find $(A \cup B) \cap C$ and how does it differ from $(A \cap B) \cup C$?

Tip:

When solving problems with sets, visualizing the sets using Venn diagrams can help make intersections and unions clearer!