Write the following sets in Roaster form :
(i) `A={x:x" is a natural number and "xlt10"}`
(ii) `B={ x : x" is a prime number which are the factor of "60}`
(iii) `C=` set of all letters of the word 'MATHEMATICS'.
(iv) `D={x:x" is an even natural number"}`
(v) `E={x:x" is an integers and "x^(2)le10}`
(vi) `F={x:x" is an integer and "-(3)/(2)ltxlt(5)/(2)"}`.

Answer

To write the given sets in roaster form, we will analyze each set based on the conditions provided.

### Solution:

**(i)** For the set \( A = \{ x : x \text{ is a natural number and } x < 10 \} \):
- Natural numbers are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).
- Therefore, in roaster form, \( A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \).

**(ii)** For the set \( B = \{ x : x \text{ is a prime number which is a factor of } 60 \} \):
- The factors of \( 60 \) are \( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \).
- The prime factors among these are \( 2, 3, 5 \).
- Therefore, in roaster form, \( B = \{ 2, 3, 5 \} \).

**(iii)** For the set \( C = \{ \text{all letters of the word 'MATHEMATICS'} \} \):
- The unique letters in 'MATHEMATICS' are \( M, A, T, H, E, I, C, S \).
- Therefore, in roaster form, \( C = \{ M, A, T, H, E, I, C, S \} \).

**(iv)** For the set \( D = \{ x : x \text{ is an even natural number} \} \):
- The even natural numbers start from \( 2 \) and continue as \( 2, 4, 6, 8, \ldots \).
- Therefore, in roaster form, we can write \( D = \{ 2, 4, 6, 8, \ldots \} \) (or specify up to a certain limit if needed).

**(v)** For the set \( E = \{ x : x \text{ is an integer and } x^2 \leq 10 \} \):
- The integers whose squares are less than or equal to \( 10 \) are \( -3, -2, -1, 0, 1, 2, 3 \).
- Therefore, in roaster form, \( E = \{ -3, -2, -1, 0, 1, 2, 3 \} \).

**(vi)** For the set \( F = \{ x : x \text{ is an integer and } -\frac{3}{2} < x < \frac{5}{2} \} \):
- The integers in this range are \( -1, 0, 1, 2 \).
- Therefore, in roaster form, \( F = \{ -1, 0, 1, 2 \} \).

### Final Answer:
- \( A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \)
- \( B = \{ 2, 3, 5 \} \)
- \( C = \{ M, A, T, H, E, I, C, S \} \)
- \( D = \{ 2, 4, 6, 8, \ldots \} \) (or specify up to a limit)
- \( E = \{ -3, -2, -1, 0, 1, 2, 3 \} \)
- \( F = \{ -1, 0, 1, 2 \} \)