If A={x,y} then power set of A is

Options

A. `{x^(y),y^(x)}`
B. `{phi,x,y}`
C. `{phi},{x},{y},{2y}`
D. `{phi},{x},{y},{x,y}`

Answer

To find the power set of the set \( A = \{x, y\} \), we will follow these steps:

### Step 1: Understand the Definition of Power Set
The power set of a set \( A \) is the set of all possible subsets of \( A \), including the empty set and \( A \) itself.

### Step 2: Determine the Number of Elements in Set \( A \)
The set \( A \) contains 2 elements: \( x \) and \( y \).

### Step 3: Use the Formula for the Number of Subsets
The number of subsets of a set with \( n \) elements is given by the formula \( 2^n \). Here, \( n = 2 \) (since \( A \) has 2 elements).

\[ \text{Number of subsets} = 2^2 = 4 \]

### Step 4: List All Possible Subsets
Now we will list all the subsets of \( A \):
1. The empty set: \( \emptyset \)
2. The set containing the first element: \( \{x\} \)
3. The set containing the second element: \( \{y\} \)
4. The set containing both elements: \( \{x, y\} \)

### Step 5: Write the Power Set
The power set \( P(A) \) is the collection of all these subsets. Therefore, we can write:

\[ P(A) = \{\emptyset, \{x\}, \{y\}, \{x, y\}\} \]

### Final Answer
Thus, the power set of \( A = \{x, y\} \) is:

\[ P(A) = \{\emptyset, \{x\}, \{y\}, \{x, y\}\} \]

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