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Let's work through the problems one by one, starting with Problem 1 on set operations. I'll provide detailed steps for each part.

### Problem 1:
We are given four sets:
- \( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- \( A = \{1, 3, 5\} \)
- \( B = \{2, 4, 5, 6, 8\} \)
- \( C = \{1, 4, 7, 8\} \)
- \( D = \{0, 3, 6, 9\} \)

#### (a) \( A \cup (C \cap B) \)
- First, compute \( C \cap B \) (intersection of sets \( C \) and \( B \)):
  \[
  C \cap B = \{4, 8\}
  \]
- Now, compute \( A \cup (C \cap B) \) (union of set \( A \) with \( C \cap B \)):
  \[
  A \cup (C \cap B) = \{1, 3, 5\} \cup \{4, 8\} = \{1, 3, 4, 5, 8\}
  \]
  
#### (b) \( (B - C) \cup (D - A) \)
- First, compute \( B - C \) (the difference of sets \( B \) and \( C \)):
  \[
  B - C = \{2, 5, 6\}
  \]
- Then, compute \( D - A \) (the difference of sets \( D \) and \( A \)):
  \[
  D - A = \{0, 6, 9\}
  \]
- Finally, compute the union \( (B - C) \cup (D - A) \):
  \[
  (B - C) \cup (D - A) = \{2, 5, 6\} \cup \{0, 6, 9\} = \{0, 2, 5, 6, 9\}
  \]

#### (c) \( (A \Delta B) - (C \Delta D) \)
- First, compute the symmetric difference \( A \Delta B \) (elements in either \( A \) or \( B \), but not both):
  \[
  A \Delta B = \{1, 2, 3, 4, 6, 8\}
  \]
- Next, compute the symmetric difference \( C \Delta D \):
  \[
  C \Delta D = \{0, 1, 3, 4, 6, 7, 8, 9\}
  \]
- Finally, compute the difference \( (A \Delta B) - (C \Delta D) \):
  \[
  (A \Delta B) - (C \Delta D) = \{2\}
  \]

#### (d) \( (D^C - A^C) \cap (B^C \cup C) \)
- First, compute the complements:
  - \( A^C = U - A = \{0, 2, 4, 6, 7, 8, 9\} \)
  - \( B^C = U - B = \{0, 1, 3, 7, 9\} \)
  - \( D^C = U - D = \{1, 2, 4, 5, 7, 8\} \)
- Compute the difference \( D^C - A^C \):
  \[
  D^C - A^C = \{1, 5\}
  \]
- Compute the union \( B^C \cup C \):
  \[
  B^C \cup C = \{0, 1, 3, 4, 7, 8, 9\}
  \]
- Finally, compute the intersection \( (D^C - A^C) \cap (B^C \cup C) \):
  \[
  (D^C - A^C) \cap (B^C \cup C) = \{1\}
  \]

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Would you like to proceed with more detailed solutions for the other problems, or explore further questions about the set operations?

### Here are some relative questions to further expand understanding:
1. What is the difference between a set union and a set intersection?
2. How does symmetric difference work in set theory?
3. What happens when we subtract one set from another in set operations?
4. Can you apply De Morgan’s law to verify set complement identities?
5. How are complements used in Venn diagrams to visualize sets?

**Tip:** To easily verify set operations, try drawing Venn diagrams! They visually represent unions, intersections, and differences.

Solve the following problems involving set theory and logic. Prove set equalities, construct truth tables, and verify logical equivalences.

Solve complex set theory problems involving unions, intersections, complements, and symmetric differences. Construct truth tables and verify logical equivalences in compound propositions. Detailed solutions with key theorems like De Morgan's Law.

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