Write the following in symbolical form (set-builder form) :
(i) Set of all even natural numbers
(ii) Set of roots of the equation `3x^(2)-5x-2=0`
(iii) Set of all positive integers greater than 3 and smaller than 8
(iv) Set of alphabets of the word 'ENGLISH'.

Answer

Let's solve the given question step by step.

### Question:
Write the following in symbolic form (set-builder form):
1. Set of all even natural numbers
2. Set of roots of the equation \(3x^{2}-5x-2=0\)
3. Set of all positive integers greater than 3 and smaller than 8
4. Set of alphabets of the word 'ENGLISH'

### Step-by-Step Solutions:

#### (i) Set of all even natural numbers
1. **Identify the elements**: We need to define the set of all even natural numbers.
2. **Define the variable**: Let \( x \) be an element of the set.
3. **Set the condition**: The condition for \( x \) is that it must be an even natural number.
4. **Write in set-builder form**:
\[ \{ x \mid x \text{ is an even natural number} \} \]

#### (ii) Set of roots of the equation \(3x^{2}-5x-2=0\)
1. **Identify the elements**: We need to find the roots of the given quadratic equation.
2. **Define the variable**: Let \( x \) be an element of the set.
3. **Set the condition**: The condition for \( x \) is that it must be a root of the equation.
4. **Write in set-builder form**:
\[ \{ x \mid x \text{ is a root of } 3x^{2}-5x-2=0 \} \]

#### (iii) Set of all positive integers greater than 3 and smaller than 8
1. **Identify the elements**: We need to define the set of positive integers that meet the criteria.
2. **Define the variable**: Let \( x \) be an element of the set.
3. **Set the conditions**: The conditions for \( x \) are that it must be a positive integer, greater than 3, and smaller than 8.
4. **Write in set-builder form**:
\[ \{ x \mid x \text{ is a positive integer and } 3 < x < 8 \} \]

#### (iv) Set of alphabets of the word 'ENGLISH'
1. **Identify the elements**: We need to define the set of letters in the word 'ENGLISH'.
2. **Define the variable**: Let \( x \) be an element of the set.
3. **Set the condition**: The condition for \( x \) is that it must be a letter in the word 'ENGLISH'.
4. **Write in set-builder form**:
\[ \{ x \mid x \text{ is a letter of the word 'ENGLISH'} \} \]

### Final Answers:
1. \(\{ x \mid x \text{ is an even natural number} \}\)
2. \(\{ x \mid x \text{ is a root of } 3x^{2}-5x-2=0 \}\)
3. \(\{ x \mid x \text{ is a positive integer and } 3 < x < 8 \}\)
4. \(\{ x \mid x \text{ is a letter of the word 'ENGLISH'} \}\)