We are tasked with determining if the given equation, $(A' \cup B)'$, is equivalent to $A' \cap B$, using the Venn diagram.

1. Expression 1: $(A' \cup B)'$

   • $A'$ refers to the complement of set $A$, which is everything outside of set $A$.
   • $A' \cup B$ means the union of the complement of $A$ and set $B$, which covers everything that is either outside of $A$ or inside $B$.
   • Taking the complement of this, $(A' \cup B)'$, means we take everything outside this union, i.e., the region that is inside $A$ but not in $B$.

2. Expression 2: $A' \cap B$

   • This refers to the intersection of the complement of $A$ and set $B$, which would cover the region that is outside of set $A$ but inside $B$.

Analysis Using the Venn Diagram:

• $(A' \cup B)'$ refers to the region inside $A$ that does not overlap with $B$ (i.e., only the leftmost part of $A$).
• $A' \cap B$ refers to the part of $B$ that is outside of $A$ (i.e., only the rightmost part of $B$).

Since these two expressions represent different regions of the Venn diagram, the two are not equal.

Conclusion:

The correct answer is False.

Would you like further details or clarifications on this? Here are some related questions:

1. What are De Morgan's laws, and how do they relate to set theory?
2. How do you apply Venn diagrams to prove set identities?
3. Can you explain how to compute the complement of a set in a Venn diagram?
4. What is the difference between union and intersection in set theory?
5. How can you simplify complex set expressions using Venn diagrams?

Tip: Always visualize set operations using a Venn diagram for clarity, especially with complements and unions.