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The problem you're dealing with involves set theory, specifically with the operations of union (U), intersection (∩), complement (Aᶜ), and set difference. The sets given are:

- A = \{-1, 2, 5\}
- B = \{-2, 0, 2, 3\}
- C = \{-1, 0, 4, 2\}

To solve the expressions, we need to define the universal set, as complements (Aᶜ, Bᶜ) are relative to that. Let's assume the universal set (U) includes all elements appearing in A, B, and C, so:

- U = \{-2, -1, 0, 2, 3, 4, 5\}

Now let's go through the given problems one by one:

### 1. \( A^c \cup B^c \) (Complement of A union Complement of B)
- First, calculate Aᶜ and Bᶜ:
  - Aᶜ = U - A = \{-2, 0, 3, 4\}
  - Bᶜ = U - B = \{-1, 4, 5\}

- Now, \( A^c \cup B^c \):
  - Aᶜ ∪ Bᶜ = \{-2, 0, 3, 4\} ∪ \{-1, 4, 5\} = \{-2, -1, 0, 3, 4, 5\}

### 2. \( (A \cap B)^c \) (Complement of the intersection of A and B)
- First, calculate \( A \cap B \):
  - A ∩ B = \{2\}

- Now, the complement of A ∩ B:
  - \( (A ∩ B)^c = U - \{2\} = \{-2, -1, 0, 3, 4, 5\} \)

### 3. \( (A - C) \cap (B \cup C) \) (Set difference of A and C intersecting with the union of B and C)
- First, calculate A - C:
  - A - C = \{5\} (since 5 is in A but not in C)

- Next, calculate B ∪ C:
  - B ∪ C = \{-2, -1, 0, 2, 3, 4\}

- Finally, intersect A - C with B ∪ C:
  - \{5\} ∩ \{-2, -1, 0, 2, 3, 4\} = ∅ (no common elements)

### 4. \( (A \cup C) - B \) (Set difference between the union of A and C and B)
- First, calculate A ∪ C:
  - A ∪ C = \{-1, 0, 2, 4, 5\}

- Now, subtract B from A ∪ C:
  - \( (A ∪ C) - B = \{-1, 0, 2, 4, 5\} - \{-2, 0, 2, 3\} = \{-1, 4, 5\} \)

### Summary of Solutions:
1. \( A^c \cup B^c = \{-2, -1, 0, 3, 4, 5\} \)
2. \( (A \cap B)^c = \{-2, -1, 0, 3, 4, 5\} \)
3. \( (A - C) \cap (B \cup C) = ∅ \)
4. \( (A \cup C) - B = \{-1, 4, 5\} \)

Would you like more details or explanations on any part?

Here are five related questions:
1. What is the complement of a set, and how is it computed?
2. How does the union of two sets differ from their intersection?
3. Can you explain how set difference works in terms of subsets?
4. What would the result of \( A \cup B \cup C \) be for these sets?
5. How does De Morgan's law apply to set complements?

**Tip:** When dealing with complements in set theory, always carefully define the universal set to ensure correct calculations.

Given the sets A = {-1, 2, 5}, B = {-2, 0, 2, 3}, and C = {-1, 0, 4, 2}, solve the following: 1. Ac U Bc 2. (A ∩ B)c 3. (A - C) ∩ (B U C) 4. (A U C) - B

This page provides a step-by-step solution to a set theory problem involving union, intersection, complements, and set difference. The sets A = {-1, 2, 5}, B = {-2, 0, 2, 3}, and C = {-1, 0, 4, 2} are analyzed using key theorems like De Morgan's Law. Perfect for students in Grades 9-12.

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