https://file.aiquickdraw.com/mathbot/file/1726201908647ro9nb5ca.jpg

Let's work through this set operations problem step by step. We are given several sets and asked to compute specific expressions involving unions, intersections, and complements.

The universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), and the subsets are:
- \( A = \{2, 3, 4, 7, 9\} \)
- \( B = \{1, 2, 3, 6, 7, 9\} \)
- \( C = \{2, 4, 6, 8\} \)
- \( D = \{1, 2, 3\} \)

We need to compute the elements of the sets for the following:

### a. \((B \cup C) \cap (\overline{B} \cap C)\)
1. First, find \( B \cup C \) (union of \( B \) and \( C \)):
   \[
   B \cup C = \{1, 2, 3, 4, 6, 7, 8, 9\}
   \]

2. Find \( \overline{B} \) (complement of \( B \) in \( U \)):
   \[
   \overline{B} = U - B = \{4, 5, 8, 10\}
   \]

3. Now, find \( \overline{B} \cap C \) (intersection of \( \overline{B} \) and \( C \)):
   \[
   \overline{B} \cap C = \{4, 8\}
   \]

4. Finally, compute \((B \cup C) \cap (\overline{B} \cap C)\):
   \[
   (B \cup C) \cap \{4, 8\} = \{4, 8\}
   \]

Thus, the result for part (a) is \( \{4, 8\} \).

### b. \( \overline{C} - [(B \cap C) \cup (A \cap C)] \)
1. First, find \( \overline{C} \) (complement of \( C \) in \( U \)):
   \[
   \overline{C} = U - C = \{1, 3, 5, 7, 9, 10\}
   \]

2. Find \( B \cap C \) (intersection of \( B \) and \( C \)):
   \[
   B \cap C = \{2, 6\}
   \]

3. Find \( A \cap C \) (intersection of \( A \) and \( C \)):
   \[
   A \cap C = \{2, 4\}
   \]

4. Compute \( (B \cap C) \cup (A \cap C) \) (union of \( B \cap C \) and \( A \cap C \)):
   \[
   (B \cap C) \cup (A \cap C) = \{2, 4, 6\}
   \]

5. Finally, compute \( \overline{C} - [(B \cap C) \cup (A \cap C)] \):
   \[
   \{1, 3, 5, 7, 9, 10\} - \{2, 4, 6\} = \{1, 3, 5, 7, 9, 10\}
   \]

Thus, the result for part (b) is \( \{1, 3, 5, 7, 9, 10\} \).

### c. \( (U \cap \overline{B}) - [\overline{A} - (A \cap C)] \)
1. First, find \( U \cap \overline{B} \):
   \[
   U \cap \overline{B} = \overline{B} = \{4, 5, 8, 10\}
   \]

2. Find \( \overline{A} \) (complement of \( A \) in \( U \)):
   \[
   \overline{A} = U - A = \{1, 5, 6, 8, 10\}
   \]

3. Find \( A \cap C \):
   \[
   A \cap C = \{2, 4\}
   \]

4. Compute \( \overline{A} - (A \cap C) \):
   \[
   \{1, 5, 6, 8, 10\} - \{2, 4\} = \{1, 5, 6, 8, 10\}
   \]

5. Finally, compute \( (U \cap \overline{B}) - [\overline{A} - (A \cap C)] \):
   \[
   \{4, 5, 8, 10\} - \{1, 5, 6, 8, 10\} = \{4\}
   \]

Thus, the result for part (c) is \( \{4\} \).

### Summary of Results:
- **a.** \( \{4, 8\} \)
- **b.** \( \{1, 3, 5, 7, 9, 10\} \)
- **c.** \( \{4\} \)

Let me know if you'd like further details or clarifications! 

Here are 5 questions that build on this:

1. How do we calculate the union and intersection of two sets?
2. What is the complement of a set and how is it determined?
3. How do we simplify set expressions that involve both complements and intersections?
4. What is the importance of the universal set \( U \) in problems like these?
5. How would the process change if \( U \) were infinite?

**Tip:** In set theory, it's crucial to clearly distinguish between union (\( \cup \)) and intersection (\( \cap \)) to avoid mistakes when calculating combined sets.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 3, 4, 7, 9}, B = {1, 2, 3, 6, 7, 9}, C = {2, 4, 6, 8}, D = {1, 2, 3}, list the elements of the sets corresponding to the following events: a. (B ∪ C) ∩ (B̅ ∩ C), b. C̅ - [(B ∩ C) ∪ (A ∩ C)], c. (U ∩ B̅) - [A̅ - (A ∩ C)]

Solve set theory problems involving the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and subsets A, B, C, and D. Learn how to compute unions, intersections, and complements step by step. This exercise is ideal for students in Grades 10-12.

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