Let's work through each part systematically to determine the elements of each set expression, given the universal set $U = \{1, 2, 3, \dots, 10\}$ and the sets $A = \{1, 2, 3, 4, 5\}$, $B = \{1, 2, 4, 8\}$, $C = \{1, 2, 3, 5, 7\}$, and $D = \{2, 4, 6, 8\}$.

Step-by-Step Solutions

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

1. Find $A \cup B = \{1, 2, 3, 4, 5, 8\}$.
2. Find the intersection with $C$: $(A \cup B) \cap C = \{1, 2, 3, 5\}$.

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

1. Find $B \cap C = \{1, 2\}$.
2. Find the union with $A$: $A \cup (B \cap C) = \{1, 2, 3, 4, 5\}$.

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

1. Find the complement of $C$ with respect to $U$, $\overline{C} = \{4, 6, 8, 9, 10\}$.
2. Find the union with $D$: $\overline{C} \cup D = \{2, 4, 6, 8, 9, 10\}$.

(d) $C \cap D$

1. Find the intersection of $C$ and $D$: $C \cap D = \{2\}$.

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

1. Find $A \cup B = \{1, 2, 3, 4, 5, 8\}$.
2. Subtract $C$ from this union: $(A \cup B) - C = \{4, 8\}$.

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

1. Find $B - C = \{4, 8\}$.
2. Find the union with $A$: $A \cup (B - C) = \{1, 2, 3, 4, 5, 8\}$.

(g) $(B - C) - D$

1. From the previous part, $B - C = \{4, 8\}$.
2. Subtract $D$ from this result: $(B - C) - D = \emptyset$ (since both 4 and 8 are in $D$).

(h) $B - (C - D)$

1. Find $C - D = \{1, 3, 5, 7\}$.
2. Subtract $C - D$ from $B$: $B - (C - D) = \{2, 4, 8\}$.

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

1. From part (d), we know $C \cap D = \{2\}$.
2. Find $A \cup B = \{1, 2, 3, 4, 5, 8\}$.
3. Subtract $C \cap D$ from this union: $(A \cup B) - (C \cap D) = \{1, 3, 4, 5, 8\}$.

Summary of Answers

• (a) $(A \cup B) \cap C = \{1, 2, 3, 5\}$
• (b) $A \cup (B \cap C) = \{1, 2, 3, 4, 5\}$
• (c) $\overline{C} \cup D = \{2, 4, 6, 8, 9, 10\}$
• (d) $C \cap D = \{2\}$
• (e) $(A \cup B) - C = \{4, 8\}$
• (f) $A \cup (B - C) = \{1, 2, 3, 4, 5, 8\}$
• (g) $(B - C) - D = \emptyset$
• (h) $B - (C - D) = \{2, 4, 8\}$
• (i) $(A \cup B) - (C \cap D) = \{1, 3, 4, 5, 8\}$

Would you like further explanations on any of these solutions?

Follow-up Questions

1. What would be the result if we add another element to set $C$ that isn't in $D$?
2. How would the results change if we took the complement of $D$ instead of $C$?
3. What if we apply different universal sets, such as $U = \{1, 2, 3, \dots, 15\}$?
4. How can these set operations be represented visually using Venn diagrams?
5. Can we apply De Morgan's laws to simplify any of these expressions?

Tip

Remember, intersections narrow down to common elements, while unions combine all unique elements in set operations.