The set `C` is letters of the word, `PAPAYA`
Write down :
All proper subsets of C .

Answer

To find all proper subsets of the set \( C \), which consists of the letters in the word "PAPAYA", we will follow these steps:

### Step 1: Identify the Set \( C \)
The first step is to identify the unique letters in the word "PAPAYA". The letters are:
- P
- A
- Y

Since the letter A appears multiple times, we only consider it once. Therefore, the set \( C \) is:
\[ C = \{ P, A, Y \} \]

### Step 2: Understand Proper Subsets
A proper subset of a set is any subset that does not include the set itself. This means we will find all possible subsets of \( C \) and then exclude \( C \) from that list.

### Step 3: List All Subsets of \( C \)
The subsets of \( C \) can be categorized based on the number of elements they contain:

1. **Empty Set**:
- \( \emptyset \)

2. **Subsets with 1 Element**:
- \( \{ P \} \)
- \( \{ A \} \)
- \( \{ Y \} \)

3. **Subsets with 2 Elements**:
- \( \{ P, A \} \)
- \( \{ P, Y \} \)
- \( \{ A, Y \} \)

4. **Subset with 3 Elements** (which is the set itself):
- \( \{ P, A, Y \} \)

### Step 4: Exclude the Set Itself
Now, we will exclude the subset that is the set \( C \) itself:
- \( \{ P, A, Y \} \)

### Step 5: List All Proper Subsets
The proper subsets of \( C \) are:
1. \( \emptyset \)
2. \( \{ P \} \)
3. \( \{ A \} \)
4. \( \{ Y \} \)
5. \( \{ P, A \} \)
6. \( \{ P, Y \} \)
7. \( \{ A, Y \} \)

### Final Answer
The proper subsets of the set \( C \) are:
\[ \emptyset, \{ P \}, \{ A \}, \{ Y \}, \{ P, A \}, \{ P, Y \}, \{ A, Y \} \]

---