Let `A, B` be two non empty subsets of a set `P`. If `(A-B)uu(B-A) = AuuB` then which of the following is correct (`X` is universal set)

Options

A. a. `A sub B`
B. b. `A sub(X-B)`
C. c. `A=B`
D. d. `B sub A`

Explanation

To solve the problem, we need to analyze the given equation:

\[ (A - B) \cup (B - A) = A \cup B \]

where \( A \) and \( B \) are non-empty subsets of a universal set \( X \).

### Step 1: Understand the components of the equation

1. **Set Difference**:
- \( A - B \) represents the elements that are in \( A \) but not in \( B \).
- \( B - A \) represents the elements that are in \( B \) but not in \( A \).

2. **Union**:
- The left side \( (A - B) \cup (B - A) \) represents the elements that are in either \( A \) or \( B \) but not in both. This is also known as the symmetric difference of sets \( A \) and \( B \).

3. **Union of Sets**:
- The right side \( A \cup B \) represents all elements that are in \( A \), in \( B \), or in both.

### Step 2: Analyze the equation

For the equation \( (A - B) \cup (B - A) = A \cup B \) to hold true, the following must be true:

- The elements in \( A \) and \( B \) must not overlap. This means that \( A \) and \( B \) must be disjoint sets. If they have any common elements, those elements would not be included in \( (A - B) \cup (B - A) \), leading to a contradiction.

### Step 3: Check the options

We need to analyze the options given in the problem to see which one satisfies the condition of the equation.

1. **Option 1**: \( A \subseteq B \)
- If \( A \) is a subset of \( B \), then \( A - B \) would be empty, and \( B - A \) would contain elements from \( B \) that are not in \( A \). Thus, \( (A - B) \cup (B - A) \) would not equal \( A \cup B \).

2. **Option 2**: \( A \subseteq X - B \)
- If \( A \) is a subset of the complement of \( B \), then \( A \) and \( B \) are disjoint. In this case, \( A - B = A \) and \( B - A = B \). Therefore, \( (A - B) \cup (B - A) = A \cup B \).

3. **Option 3**: \( A = B \)
- If \( A \) equals \( B \), then both \( A - B \) and \( B - A \) would be empty. Thus, \( (A - B) \cup (B - A) \) would be empty, which cannot equal \( A \cup B \) unless both are empty, contradicting the non-empty condition.

4. **Option 4**: \( B \subseteq A \)
- Similar to option 1, if \( B \) is a subset of \( A \), then \( B - A \) would be empty, and \( A - B \) would contain elements from \( A \) not in \( B \). Hence, \( (A - B) \cup (B - A) \) would not equal \( A \cup B \).

### Conclusion

The only option that satisfies the equation \( (A - B) \cup (B - A) = A \cup B \) is:

**Option 2: \( A \subseteq X - B \)**