If `B cap C sube A`, then `(B-A) cap (C-A)` is equal to

Options

A. `B cap C`
B. `B cup C`
C. `phi`
D. `A^(')`

Answer

To solve the problem, we need to find the intersection of the sets \( (B - A) \) and \( (C - A) \) given that \( B \cap C \subseteq A \).

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We are given that \( B \cap C \subseteq A \). This means that all elements that are in both sets \( B \) and \( C \) are also in set \( A \).

**Hint**: Remember that the intersection of two sets contains only those elements that are present in both sets.

2. **Define the Sets**:
Let's denote:
- \( B - A \): This is the set of elements that are in \( B \) but not in \( A \).
- \( C - A \): This is the set of elements that are in \( C \) but not in \( A \).

**Hint**: The operation \( B - A \) means we are removing any elements of \( A \) from \( B \).

3. **Visualize the Sets**:
Imagine sets \( A \), \( B \), and \( C \) on a Venn diagram. Since \( B \cap C \) is a subset of \( A \), any overlap between \( B \) and \( C \) is contained within \( A \). Therefore, any part of \( B \) or \( C \) that is outside of \( A \) does not overlap with the intersection.

**Hint**: Drawing a Venn diagram can help visualize the relationships between the sets.

4. **Determine \( B - A \) and \( C - A \)**:
- The set \( B - A \) consists of elements in \( B \) that are not in \( A \). Since \( B \cap C \subseteq A \), any elements of \( B \) that are also in \( A \) will be removed, leaving only those elements of \( B \) that are outside of \( A \).
- Similarly, for \( C - A \), we remove the elements of \( A \) from \( C \).

**Hint**: Focus on the elements that are exclusively in \( B \) and \( C \) and not in \( A \).

5. **Find the Intersection**:
Now we need to find \( (B - A) \cap (C - A) \). Since both \( B - A \) and \( C - A \) consist of elements that are outside of \( A \) and since \( B \cap C \subseteq A \), there are no elements that are common between \( B - A \) and \( C - A \).

**Hint**: If two sets have no elements in common, their intersection is the empty set.

6. **Conclusion**:
Therefore, the intersection \( (B - A) \cap (C - A) \) is equal to the empty set \( \phi \).

### Final Answer:
\[ (B - A) \cap (C - A) = \phi \]