If A={a,b}, find P(A).

Answer

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

### Step 1: Understand the Definition of a 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 all combinations of the elements in the original set.

### Step 2: Identify the Elements of Set A
The set \( A \) contains two elements: \( a \) and \( b \).

**Hint:** List out the elements of the set clearly to avoid confusion later.

### Step 3: List All Possible Subsets
We need to consider all combinations of the elements in \( A \):

1. The empty set: \( \emptyset \)
2. The subset containing only the first element: \( \{a\} \)
3. The subset containing only the second element: \( \{b\} \)
4. The subset containing both elements: \( \{a, b\} \)

**Hint:** Think about how many ways you can combine the elements, including choosing none.

### Step 4: Write the Power Set
Now we can compile all the subsets we found into the power set \( P(A) \):

\[ P(A) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \]

**Hint:** Ensure you include all subsets, even the empty set and the set itself.

### Step 5: Count the Number of Subsets
For a set with \( n \) elements, the power set will contain \( 2^n \) subsets. Since \( A \) has 2 elements, the power set will contain \( 2^2 = 4 \) subsets.

**Hint:** This is a quick way to verify that you have included all subsets.

### Final Answer
Thus, the power set \( P(A) \) is:

\[ P(A) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \]