The exercise you uploaded involves operations on sets such as unions, intersections, and complements based on the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$ and subsets $A, B, C, D, E$.

Here are the subsets:

• $A = \{1, 3, 5, 7, 9, 11, 13, 15\}$
• $B = \{1, 2, 7, 8, 11, 12, 15\}$
• $C = \{3, 4, 6, 7, 8, 10, 14\}$
• $D = \{2, 4, 6, 8, 10, 12, 14\}$
• $E = \{5, 6, 7, 8, 10, 12, 13\}$

For the problems listed, I will solve a few examples:

1. $A \cup C$ (Union of $A$ and $C$):

   • $A \cup C = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15\}$

2. $D \cap E$ (Intersection of $D$ and $E$):

   • $D \cap E = \{6, 8, 10, 12\}$

3. $C \cup E$ (Union of $C$ and $E$):

   • $C \cup E = \{3, 4, 5, 6, 7, 8, 10, 12, 13, 14\}$

4. $A' \cap D$ (Intersection of the complement of $A$ and $D$):

   • $A' = U - A = \{2, 4, 6, 8, 10, 12, 14\}$
   • $A' \cap D = \{2, 4, 6, 8, 10, 12, 14\}$ (since $A' = D$)

5. $B \cup E'$ (Union of $B$ and the complement of $E$):

   • $E' = U - E = \{1, 2, 3, 4, 9, 11, 14, 15\}$
   • $B \cup E' = \{1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 15\}$

Let me know if you'd like more details or the remaining exercises solved.

Here are five related questions:

1. How do we find the complement of a set?
2. Can you explain the difference between union and intersection operations?
3. What does the result of a union tell us about the two sets?
4. What does it mean if $A \cap B$ is empty?
5. How do we handle double complements in set theory?

Tip: When working with set complements, always subtract the set elements from the universal set to find the complement.