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

Options

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

Explanation

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 is the set of all possible subsets of that set, including the empty set and the set itself.

**Hint:** Remember that the power set includes every possible combination of elements from the original set.

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

**Hint:** Count the number of elements in the set to use the formula for the number of subsets.

### Step 3: Use the Formula to Find the Number of Subsets
The number of subsets of a set with \( p \) elements is given by the formula \( 2^p \). Here, \( p = 2 \) (since there are two elements in set \( A \)).

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

**Hint:** The formula \( 2^p \) helps you calculate how many subsets can be formed from the elements.

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

**Hint:** Consider combinations of including or excluding each element to form subsets.

### Step 5: Write the Power Set
The power set \( P(A) \) is the collection of all these subsets:

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

**Hint:** The power set is itself a set that contains all the subsets you listed.

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

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