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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.

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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.

Which region(s) is/are represented by A' ∪ B'? Which region(s) is/are represented by B' ∪ A'? Based on parts (a) and (b), are A' ∪ B' and B' ∪ A' equal for all sets A and B?

Explore how set theory is applied to Venn diagrams with questions on the union and complement of sets. This problem addresses the regions represented by A' ∪ B and B' ∪ A in a Venn diagram, helping students in Grades 9-12 grasp these concepts.

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