Let's solve each set operation step by step.

1. $A \cap B$ (Intersection of sets A and B): The intersection consists of elements common to both sets A and B.

   • $A = \{2, 3, 4, 6, 7\}$
   • $B = \{2, 3, 7, 8\}$
   • Common elements: $\{2, 3, 7\}$

   $A \cap B = \{2, 3, 7\}$

2. $A \cup B$ (Union of sets A and B): The union consists of all elements from both sets A and B, without repetition.

   • $A = \{2, 3, 4, 6, 7\}$
   • $B = \{2, 3, 7, 8\}$
   • All elements: $\{2, 3, 4, 6, 7, 8\}$

   $A \cup B = \{2, 3, 4, 6, 7, 8\}$

3. $(B \cup C) \cap A$ (Intersection of A with the union of B and C): First, find the union of B and C:

   • $B = \{2, 3, 7, 8\}$
   • $C = \{0, 1, 2, 4, 5, 8\}$
   • Union of B and C: $\{0, 1, 2, 3, 4, 5, 7, 8\}$

   Now, find the intersection of $A$ with $B \cup C$:

   • $A = \{2, 3, 4, 6, 7\}$
   • Common elements: $\{2, 3, 4, 7\}$

   $(B \cup C) \cap A = \{2, 3, 4, 7\}$

4. $B \cup (C \cap A)$ (Union of B with the intersection of C and A): First, find the intersection of C and A:

   • $C = \{0, 1, 2, 4, 5, 8\}$
   • $A = \{2, 3, 4, 6, 7\}$
   • Common elements: $\{2, 4\}$

   Now, find the union of B with $C \cap A$:

   • $B = \{2, 3, 7, 8\}$
   • $C \cap A = \{2, 4\}$
   • Union: $\{2, 3, 4, 7, 8\}$

   $B \cup (C \cap A) = \{2, 3, 4, 7, 8\}$

Final answers:

• $A \cap B = \{2, 3, 7\}$
• $A \cup B = \{2, 3, 4, 6, 7, 8\}$
• $(B \cup C) \cap A = \{2, 3, 4, 7\}$
• $B \cup (C \cap A) = \{2, 3, 4, 7, 8\}$

Do you want further details or explanations on any step?

Here are five related questions to deepen your understanding:

1. What is the difference between the union and intersection of sets?
2. How would you compute $A \cap (B \cup C)$?
3. How can Venn diagrams help visualize these set operations?
4. What happens if you find $A \cap (B \cap C)$?
5. How would you represent the complement of set A relative to the universal set?

Tip: The intersection of sets is always a subset of each set involved in the operation.