If A and B are any two sets, then what is `A nn (A uu B)` equal to?

Options

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

Answer

To solve the problem of finding \( A \cap (A \cup B) \), we will follow these steps:

### Step 1: Understand the Union Operation
The union of two sets \( A \) and \( B \), denoted as \( A \cup B \), is the set that contains all elements that are in \( A \), in \( B \), or in both.

### Step 2: Calculate \( A \cup B \)
Let's assume:
- \( A = \{1, 2, 3\} \)
- \( B = \{A, B, C\} \)

Now, we can find \( A \cup B \):
\[ A \cup B = \{1, 2, 3\} \cup \{A, B, C\} = \{1, 2, 3, A, B, C\} \]

### Step 3: Understand the Intersection Operation
The intersection of two sets \( A \) and \( C \), denoted as \( A \cap C \), is the set that contains all elements that are common to both sets.

### Step 4: Calculate \( A \cap (A \cup B) \)
Now we need to find \( A \cap (A \cup B) \):
\[ A \cap (A \cup B) = \{1, 2, 3\} \cap \{1, 2, 3, A, B, C\} \]
The elements that are common in both sets are \( 1, 2, 3 \).

### Step 5: Conclusion
Thus, we find that:
\[ A \cap (A \cup B) = \{1, 2, 3\} = A \]

### Final Answer
Therefore, \( A \cap (A \cup B) \) is equal to \( A \).

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