To solve the expression \( A \cap (B - C) \), we will follow these steps:
### Step 1: Understand the Sets
Let's define the sets:
- Let \( A = \{ A, B, C \} \)
- Let \( B = \{ A, B, 1 \} \)
- Let \( C = \{ 1, 2, 3 \} \)
### Step 2: Calculate \( B - C \)
The operation \( B - C \) means we need to find the elements that are in \( B \) but not in \( C \).
- Elements in \( B \): \( A, B, 1 \)
- Elements in \( C \): \( 1, 2, 3 \)
Now, we remove the elements of \( C \) from \( B \):
- \( B - C = \{ A, B \} \) (since \( 1 \) is in \( C \))
### Step 3: Calculate \( A \cap (B - C) \)
Now we need to find the intersection of \( A \) and \( (B - C) \):
- \( A = \{ A, B, C \} \)
- \( B - C = \{ A, B \} \)
The intersection \( A \cap (B - C) \) will give us the common elements:
- Common elements: \( A, B \)
Thus, \( A \cap (B - C) = \{ A, B \} \).
### Step 4: Verify with Options
Now, let's verify with the given options:
1. **Option 1**: \( A \cap B - A \cap C \)
- \( A \cap B = \{ A, B \} \)
- \( A \cap C = \emptyset \) (no common elements)
- Thus, \( A \cap B - A \cap C = \{ A, B \} - \emptyset = \{ A, B \} \) (Correct)
2. **Option 2**: \( A \cap (B - (A \cap C)) \)
- We already found \( A \cap B = \{ A, B \} \)
- \( A \cap C = \emptyset \)
- Thus, \( A \cap (B - \emptyset) = A \cap B = \{ A, B \} \) (Correct)
3. **Option 3**: \( B \cap C - A \cap C \)
- \( B \cap C = \{ 1 \} \)
- \( A \cap C = \emptyset \)
- Thus, \( B \cap C - A \cap C = \{ 1 \} - \emptyset = \{ 1 \} \) (Incorrect)
4. **Option 4**: \( B \cap C - (A \cap B \cap C) \)
- \( B \cap C = \{ 1 \} \)
- \( A \cap B \cap C = \emptyset \)
- Thus, \( B \cap C - \emptyset = \{ 1 \} \) (Incorrect)
### Conclusion
The correct expressions for \( A \cap (B - C) \) are given by options 1 and 2.
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