`A - (A - B) = ?`

Options

A. `A cup B`
B. `A cap B`
C. `B - A`
D. None of these

Answer

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

### Step 1: Understand the expression \( A - (A - B) \)

The expression \( A - (A - B) \) represents the elements that are in set \( A \) but not in the set \( A - B \).

### Step 2: Define \( A - B \)

The operation \( A - B \) means we take all the elements in set \( A \) and remove any elements that are also in set \( B \). This can be visualized as the part of set \( A \) that does not overlap with set \( B \).

### Step 3: Visualize using Venn Diagram

1. Draw two overlapping circles: one for set \( A \) and one for set \( B \).
2. The area of set \( A \) that does not overlap with set \( B \) represents \( A - B \).
3. The remaining area of set \( A \) that overlaps with set \( B \) represents \( A \cap B \) (the intersection of sets \( A \) and \( B \)).

### Step 4: Calculate \( A - (A - B) \)

Now, we need to subtract \( A - B \) from \( A \):
- When we take \( A - (A - B) \), we are left with the elements that are in \( A \) but not in the part of \( A \) that does not overlap with \( B \).
- This means we are left with the elements that are in both \( A \) and \( B \), which is \( A \cap B \).

### Conclusion

Thus, the result of the expression \( A - (A - B) \) is:
\[ A - (A - B) = A \cap B \]

### Final Answer

The answer to the expression \( A - (A - B) \) is \( A \cap B \).

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