If `X DeltaY=varphi`, then which of the following is true

Options

A. `XsubY`
B. X and Y are disjoint
C. X=Y
D. `YsubX`

Answer

To solve the problem given, we need to analyze the expression \( X \Delta Y = \varphi \) and determine which statement is true based on the properties of sets.

### Step-by-Step Solution:

1. **Understand the Symmetric Difference**:
The symmetric difference of two sets \( X \) and \( Y \), denoted as \( X \Delta Y \), is defined as:
\[ X \Delta Y = (X \cup Y) - (X \cap Y) \]
This means it includes all elements that are in either \( X \) or \( Y \) but not in both.

2. **Set the Equation**:
According to the problem, we have:
\[ X \Delta Y = \varphi \]
This means that the symmetric difference of sets \( X \) and \( Y \) is the empty set.

3. **Translate to Set Operations**:
From the definition of symmetric difference, we can rewrite the equation:
\[ \varphi = (X \cup Y) - (X \cap Y) \]
This implies that there are no elements in \( X \) or \( Y \) that are not in both sets.

4. **Conclude about the Sets**:
If \( X \Delta Y = \varphi \), it means:
\[ X \cup Y = X \cap Y \]
This can only happen if \( X \) and \( Y \) contain exactly the same elements. In other words, \( X = Y \).

5. **Example**:
To illustrate this, consider the example where \( X = \{1, 2, 3\} \) and \( Y = \{1, 2, 3\} \).
- Here, \( X \cup Y = \{1, 2, 3\} \) and \( X \cap Y = \{1, 2, 3\} \).
- Thus, \( X \Delta Y = \{1, 2, 3\} - \{1, 2, 3\} = \varphi \).

6. **Final Conclusion**:
Therefore, the correct statement is that \( X \) and \( Y \) must be equal.