If A and B are any two sets, then what is the value of `A cap (A cup B)`?

Options

A. Complement of A
B. Complement of B
C. B
D. A

Answer

To solve the problem, we need to find the value of \( A \cap (A \cup B) \).

### Step-by-Step Solution:

1. **Understand the Notation**:
- \( A \cap B \) denotes the intersection of sets A and B, which includes all elements that are in both A and B.
- \( A \cup B \) denotes the union of sets A and B, which includes all elements that are in A, in B, or in both.

2. **Apply the Distributive Law**:
- According to the distributive law of sets, we can express the intersection of A with the union of A and B as follows:
\[ A \cap (A \cup B) = (A \cap A) \cup (A \cap B) \]

3. **Simplify the Expression**:
- The intersection \( A \cap A \) is simply A, since every element in A is also in A.
- Therefore, we can rewrite the expression:
\[ A \cap (A \cup B) = A \cup (A \cap B) \]

4. **Analyze the Result**:
- The expression \( A \cup (A \cap B) \) means we take all elements from A and add any elements that are in both A and B. However, since all elements of \( A \cap B \) are already included in A, this simplifies to just A.
- Thus, we conclude:
\[ A \cap (A \cup B) = A \]

### Final Answer:
The value of \( A \cap (A \cup B) \) is \( A \).

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