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The given equation is:

\[
(A \cup B')' = A' \cap B
\]

This equation represents a logical statement involving set operations, specifically unions, complements, and intersections. Let's break it down to determine if it's true or false.

### Left-hand side: \((A \cup B')'\)

- \(A \cup B'\) means the union of set \(A\) and the complement of set \(B\).
- \((A \cup B')'\) refers to the complement of the union of \(A\) and \(B'\). By De Morgan's Law, the complement of a union is the intersection of the complements:
  \[
  (A \cup B')' = A' \cap (B'') = A' \cap B
  \]
  (Since \(B'' = B\)).

### Right-hand side: \(A' \cap B\)

This is exactly the same expression as the simplified left-hand side: \(A' \cap B\).

### Conclusion:
Both sides of the equation are identical. Therefore, the statement is **true**.

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Would you like more details or have any other questions? Here are 5 related questions to explore this topic further:

1. What is De Morgan's law in set theory?
2. How do you find the complement of a set?
3. What is the difference between union and intersection of sets?
4. Can De Morgan's law be applied to more than two sets?
5. How do Venn diagrams visually represent set operations?

**Tip:** When dealing with set operations, using Venn diagrams can help visualize complex relationships and verify results.

Understand the set theory equation (A ∪ B')' = A' ∩ B using De Morgan's Law. This step-by-step solution covers concepts of union, complement, and intersection, suitable for high school students in Grades 9-12.

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