Let's break down and solve the set operations indicated in the image. The expressions involve basic set operations such as union (∪), intersection (∩), and complement (Aᶜ), and we'll apply set theory rules step by step for each question.

1. $A^c \cup (B \cap C)^c$

• Apply De Morgan's Law to the second term: $(B \cap C)^c = B^c \cup C^c$
• Now the expression becomes: $A^c \cup (B^c \cup C^c)$
• By the associative property of the union: $(A^c \cup B^c) \cup C^c$
• The result is a union of the three complements.

2. $(C \cap D)^c \cup (B \cup C^c)$

• Apply De Morgan's Law to the first term: $(C \cap D)^c = C^c \cup D^c$
• Now the expression becomes: $(C^c \cup D^c) \cup (B \cup C^c)$
• By the associative property of the union, this simplifies to: $C^c \cup D^c \cup B$

3. $(A \cap B \cap D)^c \cup C$

• Apply De Morgan's Law to the first term: $(A \cap B \cap D)^c = A^c \cup B^c \cup D^c$
• Now the expression becomes: $(A^c \cup B^c \cup D^c) \cup C$
• By the associative property, this simplifies to: $A^c \cup B^c \cup D^c \cup C$

4. $A^c \cap B^c \cup D$

• This expression contains both union and intersection. By the precedence of operations, apply the intersection first: $A^c \cap B^c$ This represents the complement of both sets $A$ and $B$. Now union this with $D$: $(A^c \cap B^c) \cup D$

In summary, here are the final expressions:

1. $A^c \cup B^c \cup C^c$
2. $C^c \cup D^c \cup B$
3. $A^c \cup B^c \cup D^c \cup C$
4. $(A^c \cap B^c) \cup D$

Would you like any additional details or clarifications?

Here are five related questions for practice:

1. How would the operations change if union (∪) were replaced with intersection (∩)?
2. Can you apply De Morgan’s Law to the expression $(A \cup B)^c$?
3. What is the result of $A \cap (B \cup C^c)^c$?
4. How does the order of operations (union and intersection) affect the result in set theory?
5. What is the complement of $A \cup B \cap C$?

Tip: Always remember De Morgan's Laws when dealing with complements of unions or intersections. They simplify complex set expressions easily.