Write the following sets in the set builder from :
`{:((i), 0={0),(ii),P={2}, (iii),Q={"1,4,9,16"}):}`
`(iv),R={"2,3,5,7"},(v),S={"1,3,9,27"},(vi),T={"A,E,I,O,U"} `(vii) V={0,3,6,9}`

Answer

To convert the given sets into set builder form, we will analyze each set one by one and express them using the appropriate mathematical notation.

### Step-by-Step Solution:

1. **Set O = {0}**
- The only element in this set is 0. Therefore, we can express this in set builder form as:
\[ O = \{ x \mid x = 0 \} \]

2. **Set P = {2}**
- This set contains only the number 2. Hence, the set builder form is:
\[ P = \{ x \mid x = 2 \} \]

3. **Set Q = {1, 4, 9, 16}**
- These numbers are perfect squares (1², 2², 3², 4²). We can express this as:
\[ Q = \{ x \mid x = n^2, n \in \mathbb{N}, n \leq 4 \} \]
Alternatively, since the maximum value is 16, we can also write:
\[ Q = \{ x \mid x \leq 16 \text{ and } x \text{ is a perfect square} \} \]

4. **Set R = {2, 3, 5, 7}**
- These numbers are prime numbers less than 10. Thus, we can write:
\[ R = \{ x \mid x \text{ is a prime number and } x < 10 \} \]

5. **Set S = {1, 3, 9, 27}**
- These numbers are powers of 3 (3^0, 3^1, 3^2, 3^3). Therefore, we can express this as:
\[ S = \{ x \mid x = 3^n, n \in \mathbb{N}, n \leq 3 \} \]

6. **Set T = {A, E, I, O, U}**
- This set contains the English vowels. We can express it as:
\[ T = \{ x \mid x \text{ is a vowel in the English alphabet} \} \]

7. **Set V = {0, 3, 6, 9}**
- These numbers can be expressed as multiples of 3 (3 times 0, 1, 2, 3). Thus, we can write:
\[ V = \{ x \mid x = 3n, n \in \mathbb{N}, n \leq 3 \} \]

### Final Set Builder Forms:
1. \( O = \{ x \mid x = 0 \} \)
2. \( P = \{ x \mid x = 2 \} \)
3. \( Q = \{ x \mid x = n^2, n \in \mathbb{N}, n \leq 4 \} \)
4. \( R = \{ x \mid x \text{ is a prime number and } x < 10 \} \)
5. \( S = \{ x \mid x = 3^n, n \in \mathbb{N}, n \leq 3 \} \)
6. \( T = \{ x \mid x \text{ is a vowel in the English alphabet} \} \)
7. \( V = \{ x \mid x = 3n, n \in \mathbb{N}, n \leq 3 \} \)