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

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:

• $A' = \xi - A = \{2, 4, 6, 8, 10, 12\} - \{2, 6, 10, 12\} = \{4, 8\}$
• $B' = \xi - B = \{2, 4, 6, 8, 10, 12\} - \{4, 8\} = \{2, 6, 10, 12\}$
• $C' = \xi - C = \{2, 4, 6, 8, 10, 12\} - \{2, 4, 8, 10\} = \{6, 12\}$

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:

• $P \cup Q = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cup \{9, 16, 25, 36\} = \{8, 9, 12, 16, 20, 24, 25, 28, 32, 36, 40\}$
• $P \cap R = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cap \{8, 27\} = \{8\}$
• $P \cap Q = \{8, 12, 16, 20, 24, 28, 32, 36, 40\} \cap \{9, 16, 25, 36\} = \{16, 36\}$

Would you like a more detailed explanation of any part?

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.