Which pair of sets are equal? Choose all correct answers. 

The question asks which pairs of sets are equal. Let's analyze each pair from the provided options:

1. **\( A' \cup B' \) and \( (A \cap B)' \)**
   - **De Morgan's Law**: \( (A \cap B)' = A' \cup B' \).
   - **Conclusion**: This pair is equal.

2. **\( A' \cap B' \) and \( (A \cup B)' \)**
   - **De Morgan's Law**: \( (A \cup B)' = A' \cap B' \).
   - **Conclusion**: This pair is equal.

3. **\( A \cap B' \) and \( (A \cup B)' \)**
   - \( (A \cup B)' = A' \cap B' \), but \( A \cap B' \) is not the same as \( A' \cap B' \).
   - **Conclusion**: This pair is not equal.

4. **\( A' \cup B \) and \( (A \cap B)' \)**
   - \( (A \cap B)' = A' \cup B' \), but \( A' \cup B \) is not the same as \( A' \cup B' \).
   - **Conclusion**: This pair is not equal.

### Correct Answers:
- **Option 1**: \( A' \cup B' \) and \( (A \cap B)' \) are equal.
- **Option 2**: \( A' \cap B' \) and \( (A \cup B)' \) are equal.

Would you like further clarification or have any more questions? Here are some related questions you might find interesting:

1. How can De Morgan's Laws be derived from Venn diagrams?
2. Why is it important to understand set complements in probability theory?
3. How do unions and intersections of sets differ when considering infinite sets?
4. What are some real-life applications of set theory?
5. Can set theory be applied to solve logical problems?

**Tip:** When working with set operations, always remember De Morgan's Laws—they provide a quick way to find complements of unions and intersections!

Explore which pairs of sets are equal using De Morgan's Laws in set theory. This detailed analysis covers the concepts of set complements and operations, providing clarity on mathematical equivalences. Suitable for advanced high school students and anyone interested in deeper insights into set theory.

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