https://file.aiquickdraw.com/mathbot/file/17303565249695cgy53p1.jpg

Let's analyze and solve each question in the image:

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### Question 2
- **Universal Set**: \( \xi = \{2, 4, 6, 8, 10, 12\} \)
- **Subsets**:
  - \( A = \{2, 6, 10, 12\} \)
  - \( B = \{4, 8\} \)
  - \( C = \{2, 4, 8, 10\} \)

#### Requirements:
1. Find \( A' \) (the complement of \( A \)).
2. Find \( B' \) (the complement of \( B \)).
3. Find \( C' \) (the complement of \( C \)).

**Solution**:
- \( A' = \xi - A = \{2, 4, 6, 8, 10, 12\} - \{2, 6, 10, 12\} = \{4, 8\} \)
- \( B' = \xi - B = \{2, 4, 6, 8, 10, 12\} - \{4, 8\} = \{2, 6, 10, 12\} \)
- \( C' = \xi - C = \{2, 4, 6, 8, 10, 12\} - \{2, 4, 8, 10\} = \{6, 12\} \)

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### Question 3
- **Universal Set**: \( \xi = \{ \text{Positive numbers between 5 and 40} \} \), so \( \xi = \{6, 7, 8, \ldots, 40\} \).
- **Subsets**:
  - \( P = \{ \text{Multiples of 4 between 5 and 40} \} = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \)
  - \( Q = \{ \text{Perfect squares between 5 and 40} \} = \{9, 16, 25, 36\} \)
  - \( R = \{ \text{Cubes of numbers between 5 and 40} \} = \{8, 27\} \)

#### Requirements:
1. Find \( P \cup Q \) (union of \( P \) and \( Q \)).
2. Find \( P \cap R \) (intersection of \( P \) and \( R \)).
3. Find \( P \cap Q \) (intersection of \( P \) and \( Q \)).

**Solution**:
- \( P \cup Q = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cup \{9, 16, 25, 36\} = \{8, 9, 12, 16, 20, 24, 25, 28, 32, 36, 40\} \)
- \( P \cap R = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cap \{8, 27\} = \{8\} \)
- \( P \cap Q = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cap \{9, 16, 25, 36\} = \{16, 36\} \)

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Would you like a more detailed explanation of any part? 

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### Related Questions:
1. How do you find the complement of a set?
2. What is the difference between union and intersection in set theory?
3. How do you determine multiples of a number within a specific range?
4. What defines a perfect square and how do you find them within a range?
5. How can you find cube numbers within a specified range?

### Tip:
Always list elements clearly in universal sets and subsets to ensure accuracy in complements, unions, and intersections.

Given universal sets and subsets, find the complements and intersections as specified in the image.

Learn how to solve problems involving set complements, unions, and intersections with step-by-step solutions. This example covers set theory concepts like complements and union/intersection calculations. Ideal for students in Grades 8-10.

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