To solve $(A \cup B) \cap (B \cap C')$, let's break it down step by step:

Step 1: Define the universe ($U$), complements, and sets involved.

• $U = \{4, 5, 6, 7, 8, 9, 10, 11\}$
• $A = \{5, 7, 9\}$
• $B = \{4, 5, 8, 11\}$
• $C = \{4, 6, 10\}$
• $C' = U \setminus C = \{5, 7, 8, 9, 11\}$

Step 2: Compute $A \cup B$.

The union of $A$ and $B$ includes all elements that are in $A$ or $B$: $A \cup B = \{4, 5, 7, 8, 9, 11\}$

Step 3: Compute $B \cap C'$.

The intersection of $B$ and $C'$ includes all elements that are in both $B$ and $C'$: $B = \{4, 5, 8, 11\}, \quad C' = \{5, 7, 8, 9, 11\}$ $B \cap C' = \{5, 8, 11\}$

Step 4: Compute $(A \cup B) \cap (B \cap C')$.

The intersection of $A \cup B$ and $B \cap C'$ includes all elements that are in both sets: $A \cup B = \{4, 5, 7, 8, 9, 11\}, \quad B \cap C' = \{5, 8, 11\}$ $(A \cup B) \cap (B \cap C') = \{5, 8, 11\}$

Final Answer:

$\boxed{5, 8, 11}$

Would you like detailed steps or have additional questions? Here are 5 related questions to practice:

1. What is $(A \cap B) \cup (A \cap C')$?
2. Find $C \cap B'$, where $B' = U \setminus B$.
3. Compute $(A' \cup C) \cap B$.
4. How does the result of $(B \cap C)$ compare with $(B \cap C')$?
5. If $D = \{6, 9, 10, 11\}$, find $(A \cup D) \cap (B \cup C)$.

Tip: Use Venn diagrams to visually confirm set operations!