https://file.aiquickdraw.com/mathbot/file/1728873213201ra99t6au.jpg

https://file.aiquickdraw.com/mathbot/file/17288732132038r9o87e2.jpg

https://file.aiquickdraw.com/mathbot/file/17288732132036y1tv76n.jpg

Let's start by analyzing the set operations shown in the images, based on the provided sets.

We are given the universal set \( U \) and the following sets:

- \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\} \)
- \( A = \{1, 3, 5, 7, 9, 11, 13, 15\} \)
- \( B = \{1, 2, 7, 8, 11, 12, 15\} \)
- \( C = \{3, 4, 6, 7, 8, 10, 14\} \)
- \( D = \{2, 4, 6, 8, 10, 12, 14\} \)
- \( E = \{5, 6, 7, 8, 10, 12, 13\} \)

The questions involve set operations like union (\( \cup \)), intersection (\( \cap \)), difference (\( - \)), and complement (\( A' \)).

### Solving the first few exercises:

1) \( A \cup C \)
   - Union combines all elements in \( A \) or \( C \), without repetition.
   - \( A \cup C = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15\} \)

2) \( D \cap B \)
   - Intersection takes only the elements common to both sets.
   - \( D \cap B = \{2, 8, 12\} \)

3) \( C \cup E \)
   - Union combines all elements in \( C \) or \( E \).
   - \( C \cup E = \{3, 4, 5, 6, 7, 8, 10, 12, 13, 14\} \)

4) \( A' \cap D \)
   - \( A' \) is the complement of \( A \), so all elements in \( U \) but not in \( A \).
   - \( A' = \{2, 4, 6, 8, 10, 12, 14\} \)
   - \( A' \cap D = \{2, 4, 6, 8, 10, 12, 14\} \) (since \( A' \) and \( D \) are identical)

5) \( B \cup E' \)
   - First, find \( E' \), the complement of \( E \): \( E' = \{1, 2, 3, 4, 9, 11, 14, 15\} \)
   - \( B \cup E' = \{1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 15\} \)

Do you want to proceed with the next questions?

Here are 5 related questions to deepen understanding:

1. What is the result of \( A \cap D \)?
2. How would you calculate \( B \cap E' \)?
3. What is the difference between \( A - B \)?
4. Can you explain the result of \( A' \cup C \)?
5. How does set complement work in this context?

**Tip:** Always double-check each set element for accuracy in operations like union and intersection.

Perform set operations with the sets A, B, C, D, and E based on the universal set U.

This problem involves performing set operations such as union, intersection, difference, and complement on sets A, B, C, D, and E from the universal set U. It covers fundamental set theory concepts including De Morgan's laws and properties of set operations, suitable for high school students.

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