If `X` and `Y` are two sets, then `Xnn(YuuX)'` equals

Options

A. `X`
B. `Y`
C. `phi`
D. none of these

Answer

To solve the problem, we need to find the value of \( X \cap (Y \cup X') \), where \( X' \) denotes the complement of set \( X \).

### Step-by-Step Solution:

1. **Understanding the Sets**:
- Let \( X \) and \( Y \) be two sets.
- \( X' \) (the complement of \( X \)) consists of all elements that are not in \( X \).

2. **Finding \( Y \cup X' \)**:
- The union \( Y \cup X' \) includes all elements that are either in \( Y \) or not in \( X \) (or both).
- This means that \( Y \cup X' \) covers all elements in \( Y \) and all elements outside of \( X \).

3. **Finding \( X \cap (Y \cup X') \)**:
- Now we need to find the intersection of \( X \) with \( Y \cup X' \).
- The intersection \( X \cap (Y \cup X') \) includes all elements that are in \( X \) and also in \( Y \cup X' \).
- Since \( Y \cup X' \) includes all elements not in \( X \), the only elements that can be in both \( X \) and \( Y \cup X' \) are those that are in \( X \) but not in \( Y \).

4. **Analyzing the Intersection**:
- If an element is in \( X \) and also in \( Y \), it cannot be in \( X' \) (as \( X' \) contains elements not in \( X \)).
- Therefore, the intersection \( X \cap (Y \cup X') \) will only include elements that are in \( X \) but not in \( Y \).

5. **Conclusion**:
- The intersection \( X \cap (Y \cup X') \) can be simplified to \( X \setminus Y \) (elements in \( X \) that are not in \( Y \)).
- However, if there are no elements in \( X \) that are also in \( Y \), then the intersection is empty, which is denoted by \( \emptyset \) or \( \phi \).

Thus, the final answer is:
\[ X \cap (Y \cup X') = \phi \]