Consider the following statements : For any two sets A and B
(1)(A-B) `uu`B=A
(2) (A-B)`uu` A=A
(3) (A-B) `nnB=phi`
(4) `AsubeBrArrAuuB=B` of these statements:

Options

A. 1, 2, 3 are correct
B. 2, 3, 4 are correct
C. 1,3, 4 are correct
D. 1,2,4 are correct

Answer

To solve the problem, we need to analyze each of the given statements about sets A and B. Let's go through them step by step.

### Step 1: Analyze Statement 1
**Statement 1:** \( (A - B) \cup B = A \)

- **Definition of \( A - B \):** This is the set of elements that are in A but not in B.
- **Example:** Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- **Calculate \( A - B \):**
\[ A - B = \{1, 2\} \]
- **Union with B:**
\[ (A - B) \cup B = \{1, 2\} \cup \{3, 4, 5\} = \{1, 2, 3, 4, 5\} \]
- **Comparison:**
\[ \{1, 2, 3, 4, 5\} \neq A \text{ (which is } \{1, 2, 3\} \text{)} \]
- **Conclusion:** Statement 1 is **incorrect**.

### Step 2: Analyze Statement 2
**Statement 2:** \( (A - B) \cup A = A \)

- **Using the same sets:** \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- **Calculate \( A - B \):**
\[ A - B = \{1, 2\} \]
- **Union with A:**
\[ (A - B) \cup A = \{1, 2\} \cup \{1, 2, 3\} = \{1, 2, 3\} \]
- **Comparison:**
\[ \{1, 2, 3\} = A \]
- **Conclusion:** Statement 2 is **correct**.

### Step 3: Analyze Statement 3
**Statement 3:** \( (A - B) \cap B = \phi \)

- **Using the same sets:** \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- **Calculate \( A - B \):**
\[ A - B = \{1, 2\} \]
- **Intersection with B:**
\[ (A - B) \cap B = \{1, 2\} \cap \{3, 4, 5\} = \phi \]
- **Conclusion:** Statement 3 is **correct**.

### Step 4: Analyze Statement 4
**Statement 4:** If \( A \subset B \), then \( A \cup B = B \)

- **Using the same sets:** Let \( A = \{1, 2, 3\} \) and \( B = \{1, 2, 3, 4, 5, 6\} \).
- **Union of A and B:**
\[ A \cup B = \{1, 2, 3\} \cup \{1, 2, 3, 4, 5, 6\} = \{1, 2, 3, 4, 5, 6\} \]
- **Comparison:**
\[ \{1, 2, 3, 4, 5, 6\} = B \]
- **Conclusion:** Statement 4 is **correct**.

### Final Conclusion
- Statements 2, 3, and 4 are correct.
- Therefore, the answer is that statements 2, 3, and 4 are correct.

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