https://file.aiquickdraw.com/mathbot/file/1730928929348lorx7krv.jpg

To solve each of the sets described in question 6, let’s use the given sets:

- **Universal set \( U \)**: \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
- **Set \( A \)**: \(\{1, 2, 3, 4, 5\}\)
- **Set \( B \)**: \(\{1, 2, 4, 8\}\)
- **Set \( C \)**: \(\{1, 2, 3, 5, 7\}\)
- **Set \( D \)**: \(\{2, 4, 6, 8\}\)

Now we will go through each part step-by-step:

### (a) \((A \cup B) \cap C\)
1. Find \( A \cup B \): \(\{1, 2, 3, 4, 5, 8\}\)
2. Find \((A \cup B) \cap C\): Intersection with \( C = \{1, 2, 3, 5, 7\}\) gives \(\{1, 2, 3, 5\}\).

### (b) \(A \cup (B \cap C)\)
1. Find \( B \cap C \): Intersection of \( B = \{1, 2, 4, 8\}\) and \( C = \{1, 2, 3, 5, 7\}\) gives \(\{1, 2\}\).
2. Find \( A \cup (B \cap C) \): Union with \( A = \{1, 2, 3, 4, 5\}\) gives \(\{1, 2, 3, 4, 5\}\).

### (c) \( \overline{C} \cup D\)
1. Find \( \overline{C} \): Complement of \( C \) with respect to \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), so \( \overline{C} = \{4, 6, 8, 9, 10\}\).
2. Find \( \overline{C} \cup D \): Union with \( D = \{2, 4, 6, 8\}\) gives \(\{2, 4, 6, 8, 9, 10\}\).

### (d) \(C \cap D\)
Intersection of \( C = \{1, 2, 3, 5, 7\}\) and \( D = \{2, 4, 6, 8\}\) gives \(\{2\}\).

### (e) \((A \cup B) - C\)
1. Find \( A \cup B \): As calculated in (a), \(\{1, 2, 3, 4, 5, 8\}\).
2. Find \((A \cup B) - C\): Elements in \( A \cup B \) that are not in \( C = \{1, 2, 3, 5, 7\}\) gives \(\{4, 8\}\).

### (f) \(A \cup (B - C)\)
1. Find \( B - C \): Elements in \( B = \{1, 2, 4, 8\}\) that are not in \( C = \{1, 2, 3, 5, 7\}\) gives \(\{4, 8\}\).
2. Find \( A \cup (B - C) \): Union with \( A = \{1, 2, 3, 4, 5\}\) gives \(\{1, 2, 3, 4, 5, 8\}\).

### (g) \((B - C) - D\)
1. Find \( B - C \): As calculated in (f), \(\{4, 8\}\).
2. Find \((B - C) - D\): Elements in \( B - C = \{4, 8\}\) that are not in \( D = \{2, 4, 6, 8\}\) gives \(\emptyset\).

### (h) \(B - (C - D)\)
1. Find \( C - D \): Elements in \( C = \{1, 2, 3, 5, 7\}\) that are not in \( D = \{2, 4, 6, 8\}\) gives \(\{1, 3, 5, 7\}\).
2. Find \( B - (C - D) \): Elements in \( B = \{1, 2, 4, 8\}\) that are not in \( C - D = \{1, 3, 5, 7\}\) gives \(\{2, 4, 8\}\).

### (i) \((A \cup B) - (C \cap D)\)
1. Find \( C \cap D \): As calculated in (d), \(\{2\}\).
2. Find \( (A \cup B) - (C \cap D) \): Elements in \( A \cup B = \{1, 2, 3, 4, 5, 8\}\) that are not in \( C \cap D = \{2\}\) gives \(\{1, 3, 4, 5, 8\}\).

If you have any questions or need further clarifications on any of these solutions, let me know. Here are some follow-up questions to consider:

1. Can you find the elements of \( (A \cap B) \cup (C - D) \)?
2. What is \( \overline{A \cup B} \)?
3. Determine the elements of \( (A - B) \cup (D - C) \).
4. How would the answer change for \( (A \cup B) \cap (C \cup D) \)?
5. Find the elements of \( \overline{A} \cap B \).

**Tip:** When working with set complements, make sure to define the universal set clearly to avoid confusion.

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)

Solve a set theory problem involving the universal set U and subsets A, B, C, and D, with operations like union, intersection, set difference, and complement. Find the elements of each set operation, including (A ∪ B) ∩ C, A ∪ (B ∩ C), C' ∪ D, and others. Suitable for high school students in Grades 11-12.

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