If T = {x: x is a letter in the word 'TEETH'}, find all its subsets.

Answer

To solve the problem of finding all subsets of the set \( T = \{ x : x \text{ is a letter in the word 'TEETH'} \} \), we will follow these steps:

### Step 1: Identify the unique letters in the word 'TEETH'
The word 'TEETH' consists of the letters T, E, E, T, and H. However, when forming a set, we only include unique elements. Therefore, the unique letters are:
- T
- E
- H

Thus, we can define the set \( T \) as:
\[ T = \{ T, E, H \} \]

### Step 2: Determine the number of subsets
The number of subsets of a set with \( n \) elements is given by the formula \( 2^n \). In our case, \( n = 3 \) (since there are three unique letters: T, E, H).
So, the number of subsets is:
\[ 2^3 = 8 \]

### Step 3: List all the subsets
Now we will list all the subsets of the set \( T = \{ T, E, H \} \). The subsets include:
1. The empty set: \( \emptyset \)
2. Subsets with one element:
- \( \{ T \} \)
- \( \{ E \} \)
- \( \{ H \} \)
3. Subsets with two elements:
- \( \{ T, E \} \)
- \( \{ T, H \} \)
- \( \{ E, H \} \)
4. The subset with all elements:
- \( \{ T, E, H \} \)

### Final List of Subsets
Putting it all together, the complete list of subsets of \( T \) is:
1. \( \emptyset \)
2. \( \{ T \} \)
3. \( \{ E \} \)
4. \( \{ H \} \)
5. \( \{ T, E \} \)
6. \( \{ T, H \} \)
7. \( \{ E, H \} \)
8. \( \{ T, E, H \} \)

### Summary of the Solution
The subsets of the set \( T = \{ T, E, H \} \) are:
- \( \emptyset \)
- \( \{ T \} \)
- \( \{ E \} \)
- \( \{ H \} \)
- \( \{ T, E \} \)
- \( \{ T, H \} \)
- \( \{ E, H \} \)
- \( \{ T, E, H \} \)