The expression \((A \cup B)'\) represents the **complement** of the union of sets \(A\) and \(B\). In set theory, this means it includes all the elements that are **not** in the union of \(A\) and \(B\).

Using De Morgan's laws, we can rewrite \((A \cup B)'\) as:

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

### Explanation:
- **\(A \cup B\):** The union of sets \(A\) and \(B\) contains all elements that are in either \(A\), \(B\), or both.
- **\((A \cup B)'\):** The complement of \(A \cup B\) includes all elements that are **not** in \(A \cup B\).
- **De Morgan's Law:** This law helps to simplify the expression by distributing the complement over the union, turning it into an intersection: \((A \cup B)' = A' \cap B'\).

Thus, the result is the intersection of the complements of \(A\) and \(B\), which contains all elements that are neither in \(A\) nor in \(B\).

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

### Related Questions:
1. What is the complement of a set in set theory?
2. How do De Morgan's laws apply to other set operations like intersection?
3. Can you provide an example with specific sets to illustrate \((A \cup B)'\)?
4. What is the meaning of \((A \cap B)'\) and how is it different from \((A \cup B)'\)?
5. How do you visualize set operations using Venn diagrams?

### Tip:
When using De Morgan's laws, remember that the complement of a union becomes an intersection of complements, and vice versa.

Explore how to calculate the complement of the union of sets (A ∪ B)' using De Morgan's Laws in set theory. This problem involves key concepts such as union of sets, complements, and intersections. Suitable for high school students in Grades 9-12.

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