If A and B are two sets then prove `A = A nn(A uu B).`

Options

A. A
B. B
C. `phi`
D. `A nn B`

Explanation

To prove that \( A = A \cap (A \cup B) \), we will follow a step-by-step approach.

### Step 1: Understand the Sets
We have two sets, \( A \) and \( B \). We need to analyze the expression \( A \cap (A \cup B) \).

### Step 2: Define the Union of Sets
The union of sets \( A \) and \( B \) is defined as:
\[ A \cup B = \{ x | x \in A \text{ or } x \in B \} \]
This means that \( A \cup B \) contains all elements that are in \( A \), in \( B \), or in both.

### Step 3: Define the Intersection of Sets
The intersection of sets \( A \) and \( (A \cup B) \) is defined as:
\[ A \cap (A \cup B) = \{ x | x \in A \text{ and } x \in (A \cup B) \} \]
This means that \( A \cap (A \cup B) \) contains all elements that are common to both \( A \) and \( (A \cup B) \).

### Step 4: Analyze the Intersection
Since every element \( x \) in \( A \) is also in \( A \cup B \) (by the definition of union), we can say:
\[ A \cap (A \cup B) = \{ x | x \in A \} \]
Thus, we can conclude that:
\[ A \cap (A \cup B) = A \]

### Step 5: Conclusion
Since we have shown that \( A \cap (A \cup B) = A \), we can write:
\[ A = A \cap (A \cup B) \]
This completes the proof.