https://file.aiquickdraw.com/mathbot/file/1726673093863eu453p6j.jpg

Let's analyze the provided set theory problem, which asks which of the following statements are valid:

1. **(a) \( A \cup A = A \)**:  
   This is the **idempotent law** of union. The union of a set with itself is the set itself.  
   **True.**

2. **(b) \( A \cap A = A \)**:  
   This is the **idempotent law** of intersection. The intersection of a set with itself is the set itself.  
   **True.**

3. **(c) \( A \cup \emptyset = A \)**:  
   The union of any set \( A \) with the empty set \( \emptyset \) is the set \( A \).  
   **True.**

4. **(d) \( A \cup U = U \)** (where \( U \) is the universal set):  
   The union of any set \( A \) with the universal set \( U \) is the universal set \( U \).  
   **True.**

5. **(e) \( A \cap \emptyset = \emptyset \)**:  
   The intersection of any set \( A \) with the empty set \( \emptyset \) is the empty set.  
   **True.**

6. **(f) \( A \cap U = A \)** (where \( U \) is the universal set):  
   The intersection of any set \( A \) with the universal set \( U \) is the set \( A \).  
   **True.**

7. **(g) The complement of \( \bar{A} \) (denoted \( \bar{\bar{A}} \)) is \( A \)**:  
   This is the **double complement law**. The complement of the complement of a set is the set itself.  
   **True.**

### Conclusion:
All the statements provided are valid.

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Would you like further details on any of these laws? Here are some follow-up questions:

1. What is the difference between union and intersection in set theory?
2. How do the idempotent laws apply to other operations in mathematics?
3. Can you explain the role of the universal set \( U \) in set operations?
4. What happens when we apply these set operations to infinite sets?
5. How do these properties extend to Venn diagrams?

**Tip**: In set theory, learning the basic laws like the idempotent, identity, and complement laws helps in simplifying complex problems.

Which of the following statements are valid? (a) A ∪ A = A (b) A ∩ A = A (c) A ∪ ∅ = A (d) A ∪ U = U (e) A ∩ ∅ = ∅ (f) A ∩ U = A (g) The complement of A̅ is A.

This problem covers key set theory concepts such as union, intersection, and complement laws. It validates statements using the idempotent law, identity laws, and the double complement law, ideal for students in an introductory set theory course.

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