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 \( X \cap (Y \cup X)' \), we will follow these steps:

### Step 1: Understand the Sets
We have two sets \( X \) and \( Y \). The expression involves the intersection of set \( X \) with the complement of the union of sets \( Y \) and \( X \).

### Step 2: Find \( Y \cup X \)
The union of sets \( Y \) and \( X \) is defined as:
\[ Y \cup X = \{ x \in X \} \cup \{ y \in Y \} \]
This means it includes all elements that are in either set \( Y \) or set \( X \).

### Step 3: Find the Complement of \( Y \cup X \)
The complement of the union \( (Y \cup X)' \) refers to all elements that are not in \( Y \cup X \). If we consider a universal set \( U \) that contains all possible elements, then:
\[ (Y \cup X)' = U - (Y \cup X) \]
This means it includes all elements in the universal set that are not in either \( Y \) or \( X \).

### Step 4: Find the Intersection \( X \cap (Y \cup X)' \)
Now we need to find the intersection of set \( X \) with the complement of \( Y \cup X \):
\[ X \cap (Y \cup X)' = \{ x \in X \} \cap (U - (Y \cup X)) \]
This intersection will include elements that are in set \( X \) but not in the union of sets \( Y \) and \( X \).

### Step 5: Analyze the Intersection
Since \( (Y \cup X)' \) contains elements that are not in \( Y \) or \( X \), the intersection \( X \cap (Y \cup X)' \) will yield no elements because there cannot be any element that is both in \( X \) and not in \( X \). Therefore:
\[ X \cap (Y \cup X)' = \emptyset \]

### Conclusion
The final result is that the intersection is the empty set, which can be represented as:
\[ X \cap (Y \cup X)' = \emptyset \]
In terms of cardinality, this is equivalent to \( 0 \) or \( \varnothing \).

### Final Answer
Thus, the answer is \( 0 \) or \( \varnothing \).

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