If X and Y are two subsets of the unitversal set U and X' denotes the complement of X, then `X cup (X cap Y)` is equal to

Options

A. X
B. Y
C. `phi`
D. U

Answer

To solve the problem, we need to find the expression for \( X \cup (X \cap Y) \).

### Step-by-Step Solution:

1. **Understand the Components**:
- We have two sets \( X \) and \( Y \) which are subsets of a universal set \( U \).
- \( X' \) denotes the complement of set \( X \), which includes all elements in \( U \) that are not in \( X \).
- The intersection \( X \cap Y \) includes all elements that are common to both sets \( X \) and \( Y \).

**Hint**: Remember that the union of two sets includes all elements that are in either set.

2. **Apply the Union Operation**:
- We need to find \( X \cup (X \cap Y) \).
- The union operation \( \cup \) combines all elements from both sets without duplication.

3. **Analyze the Expression**:
- The term \( X \cap Y \) represents elements that are in both \( X \) and \( Y \).
- When we take the union of \( X \) with \( X \cap Y \), we are essentially adding the elements of \( X \cap Y \) to \( X \).

4. **Simplify the Expression**:
- Since \( X \cap Y \) is a subset of \( X \) (all elements in \( X \cap Y \) are already in \( X \)), the union \( X \cup (X \cap Y) \) will not add any new elements to \( X \).
- Therefore, \( X \cup (X \cap Y) = X \).

5. **Conclusion**:
- The final result is that \( X \cup (X \cap Y) = X \).

### Final Answer:
\[ X \cup (X \cap Y) = X \]