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.