`A-B =A` iff

Options

A. `A sub B`
B. `A uu B`
C. `A nnB`
D. `A-(AnnB)`

Answer

To solve the equation \( A - B = A \) if and only if, we need to analyze what this expression means in the context of set theory.

### Step-by-Step Solution:

1. **Understanding Set Difference**:
The expression \( A - B \) represents the set of elements that are in set \( A \) but not in set \( B \). Therefore, \( A - B = A \) means that removing the elements of \( B \) from \( A \) does not change \( A \).

2. **Setting Up the Equation**:
We can rewrite the equation \( A - B = A \) as:
\[ A \setminus B = A \]
This implies that all elements of \( A \) are still present in \( A \) after removing elements of \( B \).

3. **Analyzing the Implication**:
For the equation \( A \setminus B = A \) to hold true, it must mean that there are no elements in \( B \) that are also in \( A \). In other words, \( B \) must not contain any elements of \( A \).

4. **Conclusion**:
Therefore, the condition \( A - B = A \) holds true if and only if:
\[ B \cap A = \emptyset \]
This means that sets \( A \) and \( B \) are disjoint; they have no elements in common.

### Final Answer:
The statement \( A - B = A \) if and only if \( B \cap A = \emptyset \) (i.e., \( B \) is a subset of the complement of \( A \)).