To determine if the expressions $(A \cap B) \cap C$ and $(A \cup B \cup C)'$ are equal for all sets $A$, $B$, and $C$, we can use Venn diagrams to visualize both expressions and then compare them.

1. Expression 1: $(A \cap B) \cap C$

   This expression represents the intersection of sets $A$, $B$, and $C$. In a Venn diagram:

   • We first draw the intersection of $A$ and $B$, which is the region where $A$ and $B$ overlap.
   • Then, we take the intersection of this region with $C$. This will be the part of the Venn diagram where all three sets $A$, $B$, and $C$ overlap.

2. Expression 2: $(A \cup B \cup C)'$

   This expression represents the complement of the union of sets $A$, $B$, and $C$. In a Venn diagram:

   • We first draw the union of $A$, $B$, and $C$, which includes all elements that belong to at least one of the three sets.
   • The complement of this union is everything outside of $A \cup B \cup C$, meaning the area not covered by any of the sets.

Conclusion:

• $(A \cap B) \cap C$ focuses on the region common to all three sets, while $(A \cup B \cup C)'$ focuses on the region outside all three sets.
• These two expressions clearly refer to different areas of the Venn diagram. Hence, they are not equal for all sets $A$, $B$, and $C$.

Would you like me to draw the Venn diagrams to illustrate this, or clarify further?

---

Here are 5 questions to deepen understanding:

1. How do you determine the complement of a union in set theory?
2. What is the relationship between intersection and union in Venn diagrams?
3. Can two different set operations ever yield equal results? Under what conditions?
4. How does De Morgan’s law apply to complements of unions and intersections?
5. Can $(A \cap B) \cap C$ ever be equal to $(A \cup B \cup C)'$ for specific sets?

Tip: Using Venn diagrams is a powerful way to visualize complex set operations and understand how different areas correspond to different logical expressions!