Given the expression \((A \cup B) \cap (A \cup B') = A\), let's break it down step-by-step using set theory:

### 1. Definitions:
- \( A \cup B \) represents the union of sets \( A \) and \( B \).
- \( A \cup B' \) represents the union of set \( A \) with the complement of \( B \) (where \( B' \) is the complement of \( B \)).
- The intersection symbol \(\cap\) represents the common elements between the sets.

### 2. Expression Breakdown:
We need to determine the result of \((A \cup B) \cap (A \cup B')\) and show that it equals \( A \).

#### Step 1: Expand both unions.

- \( A \cup B \) includes all elements that are in \( A \), in \( B \), or in both.
- \( A \cup B' \) includes all elements that are in \( A \), or not in \( B \).

#### Step 2: Consider the intersection.
- The intersection \((A \cup B) \cap (A \cup B')\) includes all elements that are common to both \( A \cup B \) and \( A \cup B' \).

#### Step 3: Analyze.
- Any element in \( A \) will be included in both \( A \cup B \) and \( A \cup B' \), so these elements will definitely be in the intersection.
- For any element \( x \) not in \( A \):
  - If \( x \) is in \( B \), it is in \( A \cup B \), but not in \( A \cup B' \) (since it would not be in \( B' \)).
  - If \( x \) is in \( B' \), it is in \( A \cup B' \), but not in \( A \cup B \).

Thus, no elements outside \( A \) can satisfy both \( A \cup B \) and \( A \cup B' \). Therefore, only the elements of \( A \) are included in the intersection.

### Conclusion:
\[
(A \cup B) \cap (A \cup B') = A
\]

This shows that the given expression is indeed true.

Would you like more details on this, or do you have any questions?

### Related Questions:
1. What is the distributive property of sets, and how does it apply to unions and intersections?
2. How can Venn diagrams be used to visualize the expression \((A \cup B) \cap (A \cup B') = A\)?
3. What is the significance of De Morgan's laws in set theory?
4. Can the expression be proven using a Venn diagram or algebraic approach?
5. How does the complement operation affect the union and intersection of sets?

### Tip:
When working with set theory expressions, try visualizing the sets using Venn diagrams. This often provides clear insights into relationships between the sets.

Explore the set theory expression (A ∪ B) ∩ (A ∪ B') = A with a detailed explanation and step-by-step analysis. Learn about the union, intersection, and complement operations in set theory, demonstrating the application of De Morgan's Laws. Suitable for advanced high school students and beyond.

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