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Let's solve each part step-by-step. We are given:

- Universal set \( U = \{1, 2, 3, \dots, 10\} \)
- Sets:
  - \( A = \{1, 2, 3, 4, 5\} \)
  - \( B = \{1, 2, 4, 8\} \)
  - \( C = \{1, 2, 3, 5, 7\} \)
  - \( D = \{2, 4, 6, 8\} \)

Let's determine the elements of each specified set operation:

### (a) \( (A \cup B) \cap C \)

1. \( A \cup B = \{1, 2, 3, 4, 5, 8\} \)
2. \( (A \cup B) \cap C = \{1, 2, 3, 5\} \)

**Answer:** \( \{1, 2, 3, 5\} \)

### (b) \( A \cup (B \cap C) \)

1. \( B \cap C = \{1, 2\} \)
2. \( A \cup (B \cap C) = \{1, 2, 3, 4, 5\} \)

**Answer:** \( \{1, 2, 3, 4, 5\} \)

### (c) \( \overline{C} \cup D \)

1. \( \overline{C} = U - C = \{4, 6, 8, 9, 10\} \)
2. \( \overline{C} \cup D = \{2, 4, 6, 8, 9, 10\} \)

**Answer:** \( \{2, 4, 6, 8, 9, 10\} \)

### (d) \( C \cap D \)

1. \( C \cap D = \{2\} \)

**Answer:** \( \{2\} \)

### (e) \( (A \cup B) - C \)

1. \( A \cup B = \{1, 2, 3, 4, 5, 8\} \)
2. \( (A \cup B) - C = \{4, 8\} \)

**Answer:** \( \{4, 8\} \)

### (f) \( A \cup (B - C) \)

1. \( B - C = \{4, 8\} \)
2. \( A \cup (B - C) = \{1, 2, 3, 4, 5, 8\} \)

**Answer:** \( \{1, 2, 3, 4, 5, 8\} \)

### (g) \( (B - C) - D \)

1. \( B - C = \{4, 8\} \)
2. \( (B - C) - D = \emptyset \) (since \( \{4, 8\} \subset D \))

**Answer:** \( \emptyset \)

### (h) \( B - (C - D) \)

1. \( C - D = \{1, 3, 5, 7\} \)
2. \( B - (C - D) = \{2, 4, 8\} \)

**Answer:** \( \{2, 4, 8\} \)

### (i) \( (A \cup B) - (C \cap D) \)

1. \( A \cup B = \{1, 2, 3, 4, 5, 8\} \)
2. \( C \cap D = \{2\} \)
3. \( (A \cup B) - (C \cap D) = \{1, 3, 4, 5, 8\} \)

**Answer:** \( \{1, 3, 4, 5, 8\} \)

Would you like further details or explanations on any of these solutions?

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Here are some related questions you might find helpful:

1. What is the difference between union and intersection in set operations?
2. How is the complement of a set calculated within a universal set?
3. Why does the empty set appear in set operations like in part (g)?
4. How can we verify if our answer for set differences is correct?
5. How would the results change if the universal set \( U \) were larger or smaller?

**Tip:** Always confirm each intermediate step in set operations, as a small error early can affect the entire solution.

For U = {1, 2, 3, ..., 9, 10}, let A = {1, 2, 3, 4, 5}, B = {1, 2, 4, 8}, C = {1, 2, 3, 5, 7}, and D = {2, 4, 6, 8}. Determine the elements of each of the following sets: (a) (A ∪ B) ∩ C, (b) A ∪ (B ∩ C), (c) C' ∪ D, (d) C ∩ D, (e) (A ∪ B) - C, (f) A ∪ (B - C), (g) (B - C) - D, (h) B - (C - D), (i) (A ∪ B) - (C ∩ D).

Learn how to determine the elements of sets using union, intersection, complement, and set difference operations. This exercise involves sets A, B, C, D within a universal set U = {1, 2, 3, ..., 10}. Suitable for high school students studying set theory.

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