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.