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Let's break down and solve the set operations indicated in the image. The expressions involve basic set operations such as union (∪), intersection (∩), and complement (Aᶜ), and we'll apply set theory rules step by step for each question.

### 1. \( A^c \cup (B \cap C)^c \)
- Apply De Morgan's Law to the second term:
  \[
  (B \cap C)^c = B^c \cup C^c
  \]
- Now the expression becomes:
  \[
  A^c \cup (B^c \cup C^c)
  \]
- By the associative property of the union:
  \[
  (A^c \cup B^c) \cup C^c
  \]
- The result is a union of the three complements.

### 2. \( (C \cap D)^c \cup (B \cup C^c) \)
- Apply De Morgan's Law to the first term:
  \[
  (C \cap D)^c = C^c \cup D^c
  \]
- Now the expression becomes:
  \[
  (C^c \cup D^c) \cup (B \cup C^c)
  \]
- By the associative property of the union, this simplifies to:
  \[
  C^c \cup D^c \cup B
  \]

### 3. \( (A \cap B \cap D)^c \cup C \)
- Apply De Morgan's Law to the first term:
  \[
  (A \cap B \cap D)^c = A^c \cup B^c \cup D^c
  \]
- Now the expression becomes:
  \[
  (A^c \cup B^c \cup D^c) \cup C
  \]
- By the associative property, this simplifies to:
  \[
  A^c \cup B^c \cup D^c \cup C
  \]

### 4. \( A^c \cap B^c \cup D \)
- This expression contains both union and intersection. By the precedence of operations, apply the intersection first:
  \[
  A^c \cap B^c
  \]
  This represents the complement of both sets \( A \) and \( B \). Now union this with \( D \):
  \[
  (A^c \cap B^c) \cup D
  \]

In summary, here are the final expressions:
1. \( A^c \cup B^c \cup C^c \)
2. \( C^c \cup D^c \cup B \)
3. \( A^c \cup B^c \cup D^c \cup C \)
4. \( (A^c \cap B^c) \cup D \)

Would you like any additional details or clarifications?

Here are five related questions for practice:
1. How would the operations change if union (∪) were replaced with intersection (∩)?
2. Can you apply De Morgan’s Law to the expression \( (A \cup B)^c \)?
3. What is the result of \( A \cap (B \cup C^c)^c \)?
4. How does the order of operations (union and intersection) affect the result in set theory?
5. What is the complement of \( A \cup B \cap C \)?

**Tip:** Always remember De Morgan's Laws when dealing with complements of unions or intersections. They simplify complex set expressions easily.

Perform the indicated operations: 1. A^c ∪ (B ∩ C)^c, 2. (C ∩ D)^c ∪ (B ∪ C^c), 3. (A ∩ B ∩ D)^c ∪ C, 4. A^c ∩ B^c ∪ D.

Learn how to solve set operations involving union, intersection, and complements using De Morgan's Law. Includes step-by-step solutions for problems like A^c ∪ (B ∩ C)^c and more. Suitable for high school students (Grades 9-12).

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