For any two sets A and B prove the following: `A-(A-B)=AnnB`

Explanation

To prove that \( A - (A - B) = A \cap B \) for any two sets \( A \) and \( B \), we will show that both sides of the equation are equal by demonstrating that each side is a subset of the other.

### Step 1: Show that \( A - (A - B) \subseteq A \cap B \)

1. **Let \( x \in A - (A - B) \)**: This means that \( x \) is an element of \( A \) and not an element of \( A - B \).
2. **Since \( x \in A \)**: By definition of set difference, \( x \in A - B \) means \( x \in A \) and \( x \notin B \).
3. **Since \( x \notin A - B \)**: This means \( x \) is either not in \( A \) or it is in \( B \). However, since we already know \( x \in A \), the only possibility left is that \( x \in B \).
4. **Thus, \( x \in A \) and \( x \in B \)**: Therefore, \( x \in A \cap B \).
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