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The exercise you uploaded involves operations on sets such as unions, intersections, and complements based on the universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\} \) and subsets \( A, B, C, D, E \).

Here are the subsets:

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

For the problems listed, I will solve a few examples:

1. **\( A \cup C \)** (Union of \( A \) and \( C \)):
   - \( A \cup C = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15\} \)

2. **\( D \cap E \)** (Intersection of \( D \) and \( E \)):
   - \( D \cap E = \{6, 8, 10, 12\} \)

3. **\( C \cup E \)** (Union of \( C \) and \( E \)):
   - \( C \cup E = \{3, 4, 5, 6, 7, 8, 10, 12, 13, 14\} \)

4. **\( A' \cap D \)** (Intersection of the complement of \( A \) and \( D \)):
   - \( A' = U - A = \{2, 4, 6, 8, 10, 12, 14\} \)
   - \( A' \cap D = \{2, 4, 6, 8, 10, 12, 14\} \) (since \( A' = D \))

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

Let me know if you'd like more details or the remaining exercises solved.

Here are five related questions:
1. How do we find the complement of a set?
2. Can you explain the difference between union and intersection operations?
3. What does the result of a union tell us about the two sets?
4. What does it mean if \( A \cap B \) is empty?
5. How do we handle double complements in set theory?

**Tip:** When working with set complements, always subtract the set elements from the universal set to find the complement.

The exercise uploaded involves operations on sets such as unions, intersections, complements, and differences using the sets A, B, C, D, and E.

Explore set operations with detailed solutions, including union, intersection, complement, and set differences using sets A, B, C, D, and E. Ideal for students in Grades 8-12 studying set theory.

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