Let's work through each of the set operations one by one using the universal set $U = \{1, 2, 3, 4, 5, 6, 7\}$, and the sets:

• $A = \{1, 3, 5, 7\}$
• $B = \{1, 2, 3\}$
• $C = \{2, 3, 4, 5, 6\}$

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

• First, find $B \cap C$, which is the intersection of sets $B$ and $C$. $B \cap C = \{2, 3\}$
• Now, find $A \cup (B \cap C)$, which is the union of sets $A$ and $\{2, 3\}$. $A \cup (B \cap C) = \{1, 2, 3, 5, 7\}$

2. $A \cap (B \cup C)$

• First, find $B \cup C$, which is the union of sets $B$ and $C$. $B \cup C = \{1, 2, 3, 4, 5, 6\}$
• Now, find $A \cap (B \cup C)$, which is the intersection of sets $A$ and $\{1, 2, 3, 4, 5, 6\}$. $A \cap (B \cup C) = \{1, 3, 5\}$

3. $(A \cup B) \cap (A \cup C)$

• First, find $A \cup B$, which is the union of sets $A$ and $B$. $A \cup B = \{1, 2, 3, 5, 7\}$
• Then, find $A \cup C$, which is the union of sets $A$ and $C$. $A \cup C = \{1, 2, 3, 4, 5, 6, 7\}$
• Now, find the intersection of these two sets. $(A \cup B) \cap (A \cup C) = \{1, 2, 3, 5, 7\}$

4. $(A \cap B) \cup (A \cap C)$

• First, find $A \cap B$, which is the intersection of sets $A$ and $B$. $A \cap B = \{1, 3\}$
• Then, find $A \cap C$, which is the intersection of sets $A$ and $C$. $A \cap C = \{3, 5\}$
• Now, find the union of these two sets. $(A \cap B) \cup (A \cap C) = \{1, 3, 5\}$

5. $A' \cap (B \cup C)$

• First, find $A'$, which is the complement of set $A$ with respect to $U$. $A' = U \setminus A = \{2, 4, 6\}$
• Then, find $B \cup C$. $B \cup C = \{1, 2, 3, 4, 5, 6\}$
• Now, find the intersection of these two sets. $A' \cap (B \cup C) = \{2, 4, 6\}$

Would you like me to continue solving the rest of the exercises or explain any of the steps in more detail?

Here are 5 related questions for further practice:

1. What is $(A' \cup B') \cap C$?
2. Can you find the complement of $(B \cap C)$ with respect to $U$?
3. What is the difference between $A \cap (B \cup C)$ and $(A \cap B) \cup (A \cap C)$?
4. How do you compute $A' \cap B'$?
5. What does $(A \cup B)'$ represent in terms of set operations?

Tip: Always simplify the inner parentheses first in complex set operations, especially for union and intersection.