To solve the problem, we will match the set operations with their corresponding Venn diagrams by interpreting the shaded regions.

Let's break down each set operation and match it with the appropriate diagram:

1. 4. $A \cap B \cap C$

   • This represents the intersection of all three sets, meaning the region that is common to $A$, $B$, and $C$ should be shaded. This is typically the central part where all three circles overlap.
   • Match: Diagram e (The shaded region is in the center where $A$, $B$, and $C$ overlap).

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

   • $A^C$ means the complement of $A$, or everything outside of $A$. $B \cup C$ is the union of $B$ and $C$, meaning everything inside $B$ or $C$. The intersection of these two sets is the area outside $A$ but within $B$ or $C$.
   • Match: Diagram f (Shaded region covers areas outside $A$, but inside $B$ or $C$).

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

   • $A^C$ is the complement of $A$, so the area outside of $A$. $(B \cup C)^C$ is the complement of $B \cup C$, which means everything outside both $B$ and $C$. The intersection is the area that is outside of all three sets.
   • Match: Diagram d (The entire region outside all three circles is shaded).

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

   • Let's break it down:
     • $B^C$ is the complement of $B$, or everything outside $B$.
     • $A^C \cup C^C$ is the union of the complements of $A$ and $C$, meaning everything outside $A$ or outside $C$.
     • $B^C \cap (A^C \cup C^C)$ is the region outside $B$, and either outside $A$ or outside $C$. Finally, taking the complement of this region means shading the area inside $B$ and either inside $A$ or inside $C$.
   • Match: Diagram c (Shaded region is inside $B$ and parts of $A$ and $C$).

Summary of Matches:

• 4. $A \cap B \cap C$ → Diagram e
• 5. $A^C \cap (B \cup C)$ → Diagram f
• 6. $A^C \cap (B \cup C)^C$ → Diagram d
• 7. $(B^C \cap (A^C \cup C^C))^C$ → Diagram c

Would you like any further clarification or details on these set operations?

5 Related Questions:

1. What is the difference between union and intersection in set theory?
2. How do we find the complement of a set in a Venn diagram?
3. Can you explain the meaning of $(B^C \cap (A^C \cup C^C))^C$ in more detail?
4. How would the diagram change if we used $A \cup B \cup C$ instead of the intersection?
5. What is the significance of the complement in set theory?

Tip:

When working with set operations in Venn diagrams, always visualize each component step by step, particularly for complements and unions, to avoid confusion.