Which of the following are corect: 1. `A-B=A - (Ann B)` 2. `A = (A nnB) uu (A-B)` 3.`A - (B uu C) =(A-B) uu (A-C)`

Options

A. 1 and 3
B. 2 only
C. 2 and 3
D. 1 and 2

Answer

To determine which of the given statements are correct, we will analyze each one step by step.

### Given Sets:
Let's define the sets:
- Universal set \( U = \{1, 2, 3, 4, 5, 6\} \)
- Set \( A = \{1, 2, 3, 4\} \)
- Set \( B = \{3, 4, 5\} \)
- Set \( C = \{4, 5, 6\} \)

### Statement 1: \( A - B = A - (A \cap B) \)

1. **Calculate \( A - B \)**:
- \( A - B \) means we take all elements in \( A \) that are not in \( B \).
- \( A = \{1, 2, 3, 4\} \) and \( B = \{3, 4, 5\} \).
- So, \( A - B = \{1, 2\} \).

2. **Calculate \( A \cap B \)**:
- The intersection \( A \cap B \) gives us the common elements in both sets.
- \( A \cap B = \{3, 4\} \).

3. **Calculate \( A - (A \cap B) \)**:
- Now we need to find \( A - (A \cap B) \).
- \( A - (A \cap B) = A - \{3, 4\} = \{1, 2\} \).

4. **Comparison**:
- LHS: \( A - B = \{1, 2\} \)
- RHS: \( A - (A \cap B) = \{1, 2\} \)
- Since LHS = RHS, **Statement 1 is correct**.

### Statement 2: \( A = (A \cap B) \cup (A - B) \)

1. **Calculate \( A \cap B \)**:
- From the previous calculation, \( A \cap B = \{3, 4\} \).

2. **Calculate \( A - B \)**:
- From the previous calculation, \( A - B = \{1, 2\} \).

3. **Calculate \( (A \cap B) \cup (A - B) \)**:
- Now we find the union of \( A \cap B \) and \( A - B \).
- \( (A \cap B) \cup (A - B) = \{3, 4\} \cup \{1, 2\} = \{1, 2, 3, 4\} \).

4. **Comparison**:
- LHS: \( A = \{1, 2, 3, 4\} \)
- RHS: \( (A \cap B) \cup (A - B) = \{1, 2, 3, 4\} \)
- Since LHS = RHS, **Statement 2 is correct**.

### Statement 3: \( A - (B \cup C) = (A - B) \cup (A - C) \)

1. **Calculate \( B \cup C \)**:
- \( B = \{3, 4, 5\} \) and \( C = \{4, 5, 6\} \).
- \( B \cup C = \{3, 4, 5, 6\} \).

2. **Calculate \( A - (B \cup C) \)**:
- \( A - (B \cup C) = A - \{3, 4, 5, 6\} \).
- Since \( A = \{1, 2, 3, 4\} \), we remove \( 3 \) and \( 4 \).
- Thus, \( A - (B \cup C) = \{1, 2\} \).

3. **Calculate \( A - B \)**:
- From previous calculations, \( A - B = \{1, 2\} \).

4. **Calculate \( A - C \)**:
- \( A - C = A - \{4, 5, 6\} \).
- Removing \( 4 \) from \( A \) gives us \( A - C = \{1, 2, 3\} \).

5. **Calculate \( (A - B) \cup (A - C) \)**:
- Now we find the union of \( A - B \) and \( A - C \).
- \( (A - B) \cup (A - C) = \{1, 2\} \cup \{1, 2, 3\} = \{1, 2, 3\} \).

6. **Comparison**:
- LHS: \( A - (B \cup C) = \{1, 2\} \)
- RHS: \( (A - B) \cup (A - C) = \{1, 2, 3\} \)
- Since LHS ≠ RHS, **Statement 3 is incorrect**.

### Final Conclusion:
- **Correct Statements**: 1 and 2
- **Incorrect Statement**: 3