Let's analyze the Venn diagram and the questions related to it. The sets A and B are represented in the diagram with regions labeled I, II, III, and the universal set $U$ covers all regions.

Here’s a breakdown:

1. $A' \cup B$: This represents everything that is not in set A (the complement of A, $A'$) or in set B. In terms of regions:

   • $A'$ includes regions that are not in A, which are regions III and IV.
   • $B$ includes all regions inside set B, which are regions II and III.

   So, $A' \cup B$ includes regions II, III, and IV.

2. $B' \cup A$: This represents everything that is not in set B (the complement of B, $B'$) or in set A. In terms of regions:

   • $B'$ includes regions that are not in B, which are regions I and IV.
   • $A$ includes all regions inside set A, which are regions I and II.

   So, $B' \cup A$ includes regions I, II, and IV.

Now let's answer the specific questions:

a. Which region(s) is/are represented by $A' \cup B$?

From the analysis above, $A' \cup B$ corresponds to regions II, III, IV.

b. Which region(s) is/are represented by $B' \cup A$?

$B' \cup A$ corresponds to regions I, II, IV.

c. Based on parts (a) and (b), are $A' \cup B$ and $B' \cup A$ equal for all sets A and B?

No, $A' \cup B$ and $B' \cup A$ are not equal for all sets A and B, as they cover different regions. $A' \cup B$ includes region III, while $B' \cup A$ includes region I.

Would you like further details or have any questions about this explanation?

Here are 5 related questions for further exploration:

1. What is the region represented by $A' \cap B$?
2. How can we represent $A \cup B$ in terms of regions in the Venn diagram?
3. What is the complement of $A' \cup B$, and which regions would it include?
4. How does $A' \cap B'$ differ from $A \cap B$?
5. Can we use set notation to express the symmetric difference of $A$ and $B$ in this diagram?

Tip: In Venn diagrams, the union of two sets $A \cup B$ represents all areas covered by either set, while the intersection $A \cap B$ represents only the area common to both sets.