If C is the set of letters in the word 'cooler', find:
(i) setC (ii) n ( C)
(iii) number of its subsets (iv) number of its proper subsets

Answer

To solve the problem step by step, we will address each part of the question regarding the set of letters in the word "cooler".

### Step 1: Identify the Set C
The first task is to determine the set of unique letters in the word "cooler".

- The letters in "cooler" are: c, o, o, l, e, r.
- To form a set, we only include unique elements. Therefore, we will remove the duplicate 'o'.

Thus, the set C is:
\[ C = \{ c, o, l, e, r \} \]

### Step 2: Count the Number of Elements in Set C
Next, we need to find the number of elements in set C, denoted as \( n(C) \).

- The elements in set C are: c, o, l, e, r.
- Counting these gives us 5 unique elements.

So,
\[ n(C) = 5 \]

### Step 3: Calculate the Number of Subsets
The number of subsets of a set can be calculated using the formula:
\[ \text{Number of subsets} = 2^n \]
where \( n \) is the number of elements in the set.

- Since \( n(C) = 5 \), we calculate:
\[ \text{Number of subsets} = 2^5 = 32 \]

### Step 4: Calculate the Number of Proper Subsets
Proper subsets are all the subsets of a set except the set itself. The formula for the number of proper subsets is:
\[ \text{Number of proper subsets} = 2^n - 1 \]

- Using \( n(C) = 5 \):
\[ \text{Number of proper subsets} = 2^5 - 1 = 32 - 1 = 31 \]

### Summary of Results
1. Set C: \( \{ c, o, l, e, r \} \)
2. \( n(C) = 5 \)
3. Number of subsets = 32
4. Number of proper subsets = 31