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Let's work through each of the set operations one by one using the universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \), and the sets:

- \( A = \{1, 3, 5, 7\} \)
- \( B = \{1, 2, 3\} \)
- \( C = \{2, 3, 4, 5, 6\} \)

### 1. \( A \cup (B \cap C) \)
- First, find \( B \cap C \), which is the intersection of sets \( B \) and \( C \).
  \[
  B \cap C = \{2, 3\}
  \]
- Now, find \( A \cup (B \cap C) \), which is the union of sets \( A \) and \( \{2, 3\} \).
  \[
  A \cup (B \cap C) = \{1, 2, 3, 5, 7\}
  \]

### 2. \( A \cap (B \cup C) \)
- First, find \( B \cup C \), which is the union of sets \( B \) and \( C \).
  \[
  B \cup C = \{1, 2, 3, 4, 5, 6\}
  \]
- Now, find \( A \cap (B \cup C) \), which is the intersection of sets \( A \) and \( \{1, 2, 3, 4, 5, 6\} \).
  \[
  A \cap (B \cup C) = \{1, 3, 5\}
  \]

### 3. \( (A \cup B) \cap (A \cup C) \)
- First, find \( A \cup B \), which is the union of sets \( A \) and \( B \).
  \[
  A \cup B = \{1, 2, 3, 5, 7\}
  \]
- Then, find \( A \cup C \), which is the union of sets \( A \) and \( C \).
  \[
  A \cup C = \{1, 2, 3, 4, 5, 6, 7\}
  \]
- Now, find the intersection of these two sets.
  \[
  (A \cup B) \cap (A \cup C) = \{1, 2, 3, 5, 7\}
  \]

### 4. \( (A \cap B) \cup (A \cap C) \)
- First, find \( A \cap B \), which is the intersection of sets \( A \) and \( B \).
  \[
  A \cap B = \{1, 3\}
  \]
- Then, find \( A \cap C \), which is the intersection of sets \( A \) and \( C \).
  \[
  A \cap C = \{3, 5\}
  \]
- Now, find the union of these two sets.
  \[
  (A \cap B) \cup (A \cap C) = \{1, 3, 5\}
  \]

### 5. \( A' \cap (B \cup C) \)
- First, find \( A' \), which is the complement of set \( A \) with respect to \( U \).
  \[
  A' = U \setminus A = \{2, 4, 6\}
  \]
- Then, find \( B \cup C \).
  \[
  B \cup C = \{1, 2, 3, 4, 5, 6\}
  \]
- Now, find the intersection of these two sets.
  \[
  A' \cap (B \cup C) = \{2, 4, 6\}
  \]

Would you like me to continue solving the rest of the exercises or explain any of the steps in more detail?

Here are 5 related questions for further practice:
1. What is \( (A' \cup B') \cap C \)?
2. Can you find the complement of \( (B \cap C) \) with respect to \( U \)?
3. What is the difference between \( A \cap (B \cup C) \) and \( (A \cap B) \cup (A \cap C) \)?
4. How do you compute \( A' \cap B' \)?
5. What does \( (A \cup B)' \) represent in terms of set operations?

**Tip:** Always simplify the inner parentheses first in complex set operations, especially for union and intersection.

Practice Exercises: Solve the set theory problems using sets U, A, B, and C.

This problem set covers fundamental operations in set theory, including union, intersection, and complement of sets A, B, and C. Detailed step-by-step solutions are provided for students to practice and enhance their understanding of set operations.

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