if X and Y are two sets , then `X cap (Y cup X)'`equals

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understand the Expression**:
We have the expression \( X \cap (Y \cup X)' \). The first step is to recognize that \( (Y \cup X)' \) is the complement of the union of sets \( Y \) and \( X \).

2. **Apply De Morgan's Law**:
According to De Morgan's Laws, the complement of a union can be expressed as the intersection of the complements:
\[ (Y \cup X)' = Y' \cap X' \]
Therefore, we can rewrite our expression as:
\[ X \cap (Y' \cap X') \]

3. **Rearranging the Intersection**:
The intersection operation is associative, so we can rearrange the terms:
\[ X \cap (Y' \cap X') = (X \cap Y') \cap X' \]

4. **Using the Identity of Intersection**:
The intersection of a set with its complement is always the empty set:
\[ X \cap X' = \emptyset \]
Therefore, we have:
\[ (X \cap Y') \cap X' = \emptyset \]

5. **Final Result**:
Hence, the value of the expression \( X \cap (Y \cup X)' \) is:
\[ \emptyset \]

### Conclusion:
The final answer is \( \emptyset \).