For any two sets A and `B, A-(A-B)` equals

Options

A. B
B. `A-B`
C. `A nnB`
D. `A ^(c ) nn B ^(c )`

Answer

To solve the problem \( A - (A - B) \), we will follow a step-by-step approach using set theory concepts.

### Step 1: Understand the notation
The notation \( A - B \) represents the set difference, which means the elements that are in set \( A \) but not in set \( B \).

### Step 2: Calculate \( A - B \)
Let's denote \( A - B \) as the set of elements that are in \( A \) but not in \( B \). This can be written as:
\[ A - B = \{ x \in A \mid x \notin B \} \]

### Step 3: Substitute into the original expression
Now, we need to substitute \( A - B \) into the original expression \( A - (A - B) \):
\[ A - (A - B) = A - \{ x \in A \mid x \notin B \} \]

### Step 4: Interpret \( A - (A - B) \)
The expression \( A - (A - B) \) means we are removing all elements that are in \( A - B \) from \( A \).

### Step 5: Identify the remaining elements
When we remove the elements that are in \( A - B \) from \( A \), we are left with the elements that are in \( A \) and also in \( B \). This is the intersection of sets \( A \) and \( B \):
\[ A - (A - B) = A \cap B \]

### Conclusion
Thus, we conclude that:
\[ A - (A - B) = A \cap B \]