Let `A_(1)`, `A_(2)` and `A_(3)` be subsets of a set `X`. Which one of the following is correct ?

Options

A. `A_(1)uuA_(2)uuA_(3)` is the smallest subset of `X` containing elements of each of `A_(1)`, `A_(2)` and `A_(3)`
B. `A_(1)uuA_(2)uuA_(3)` is the smallest subset of `X` containing elements either `A_(1)` or `A_(2)uuA_(3)` but not both
C. The smallest subset of `X` containing `A_(1)uuA_(2)` and `A_(3)` equals the smallest subset of `X` containing both `A_(1)` and `A_(2)uuA_(3)` only if `A_(2)=A_(3)`
D. None of these

Answer

To solve the problem, we need to analyze the statements regarding the subsets \( A_1 \), \( A_2 \), and \( A_3 \) of a set \( X \).

### Step-by-Step Solution:

1. **Understanding the Union of Sets**:
- The union of sets \( A_1 \), \( A_2 \), and \( A_3 \) is defined as:
\[ A_1 \cup A_2 \cup A_3 = \{ x \in X : x \in A_1 \text{ or } x \in A_2 \text{ or } x \in A_3 \} \]
- This means that \( A_1 \cup A_2 \cup A_3 \) contains all elements that are in at least one of the subsets \( A_1 \), \( A_2 \), or \( A_3 \).

2. **Analyzing the First Statement**:
- The first statement claims that \( A_1 \cup A_2 \cup A_3 \) is the smallest subset of \( X \) containing elements of \( A_1 \), \( A_2 \), and \( A_3 \).
- Since \( A_1 \), \( A_2 \), and \( A_3 \) are subsets of \( X \), their union indeed contains all elements from these subsets and is the smallest set that does so.
- **Conclusion**: This statement is **true**.

3. **Analyzing the Second Statement**:
- The second statement claims that \( A_1 \cup A_2 \cup A_3 \) is the smallest subset of \( X \) containing elements either from \( A_1 \) or \( A_2 \cup A_3 \) but not both.
- However, \( A_1 \cup A_2 \cup A_3 \) includes elements from both \( A_1 \) and \( A_2 \cup A_3 \). Therefore, this statement is **false**.

4. **Analyzing the Third Statement**:
- The third statement claims that the smallest subset of \( X \) containing elements of \( A_1 \cup A_2 \) and \( A_3 \) is equal to the smallest subset of \( X \) containing both \( A_1 \) and \( A_2 \cup A_3 \) only if \( A_2 = A_3 \).
- The smallest subset containing \( A_1 \cup A_2 \cup A_3 \) is \( A_1 \cup A_2 \cup A_3 \), regardless of whether \( A_2 \) equals \( A_3 \) or not.
- **Conclusion**: This statement is also **false**.

### Final Conclusion:
- The only correct statement is the first one. Therefore, the answer is:
\[ \text{Correct Statement: } A_1 \cup A_2 \cup A_3 \text{ is the smallest subset of } X \text{ containing elements of } A_1, A_2, A_3. \]