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

Options

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

Explanation

To solve the problem, we need to find the expression for \( X \cap (Y \cap X') \), where \( X' \) is the complement of set \( X \).

### Step-by-Step Solution:

1. **Understand the Expression**: We need to evaluate \( X \cap (Y \cap X') \). Here, \( X' \) denotes the complement of set \( X \).

2. **Use Associative Property**: The intersection operation is associative, which means we can rearrange the parentheses. Thus, we can write:
\[ X \cap (Y \cap X') = (X \cap Y) \cap X' \]

3. **Apply the Identity**: We know that \( A \cap B' = A' \cup B' \). Therefore, we can rewrite \( (X \cap Y) \cap X' \) using the identity:
\[ (X \cap Y) \cap X' = (X \cap Y') \cup (X \cap X') \]

4. **Evaluate \( X \cap X' \)**: The intersection of a set and its complement is always the empty set, denoted as \( \emptyset \):
\[ X \cap X' = \emptyset \]

5. **Combine Results**: Now substituting back into our expression, we have:
\[ (X \cap Y') \cup \emptyset = X \cap Y' \]

6. **Final Result**: Thus, the expression simplifies to:
\[ X \cap (Y \cap X') = X \cap Y' \]

### Conclusion:
The final answer is \( X \cap Y' \).

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