Write the following sets in the set-builder form :
(i) `A = {2,4,6,8,10}`
(ii) `B = {1,2,3,4,6,12}`
(iii) `C= {1,3,5,7,9….}`
(iv) `D={(1)/(2),(1)/(3),(1)/(4),...}`
(v) `E={2,4,8,16,32,64}`.

Answer

To express the given sets in set-builder form, we will identify the pattern or property that defines the elements of each set. Let's go through each set step by step.

### Step-by-Step Solution:

**(i) Set A = {2, 4, 6, 8, 10}**

1. Identify the elements: The elements are 2, 4, 6, 8, and 10.
2. Recognize the pattern: All elements are even natural numbers less than or equal to 10.
3. Write in set-builder form:
\[ A = \{ x \mid x \text{ is an even natural number and } x \leq 10 \} \]

**(ii) Set B = {1, 2, 3, 4, 6, 12}**

1. Identify the elements: The elements are 1, 2, 3, 4, 6, and 12.
2. Recognize the pattern: All elements are factors of 12.
3. Write in set-builder form:
\[ B = \{ x \mid x \text{ is a factor of } 12 \} \]

**(iii) Set C = {1, 3, 5, 7, 9, ...}**

1. Identify the elements: The elements are odd numbers starting from 1.
2. Recognize the pattern: All elements are odd natural numbers.
3. Write in set-builder form:
\[ C = \{ x \mid x \text{ is an odd natural number} \} \]

**(iv) Set D = \{ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \}**

1. Identify the elements: The elements are fractions of the form \(\frac{1}{n}\) where \(n\) is a natural number starting from 2.
2. Recognize the pattern: Each element can be expressed as \(\frac{1}{n}\) where \(n\) is a natural number greater than or equal to 2.
3. Write in set-builder form:
\[ D = \{ x \mid x = \frac{1}{n}, n \text{ is a natural number} \} \]

**(v) Set E = {2, 4, 8, 16, 32, 64}**

1. Identify the elements: The elements are powers of 2.
2. Recognize the pattern: Each element can be expressed as \(2^n\) where \(n\) is a natural number from 1 to 6.
3. Write in set-builder form:
\[ E = \{ x \mid x = 2^n, n \text{ is a natural number and } n \leq 6 \} \]

### Final Answers in Set-Builder Form:

- \( A = \{ x \mid x \text{ is an even natural number and } x \leq 10 \} \)
- \( B = \{ x \mid x \text{ is a factor of } 12 \} \)
- \( C = \{ x \mid x \text{ is an odd natural number} \} \)
- \( D = \{ x \mid x = \frac{1}{n}, n \text{ is a natural number} \} \)
- \( E = \{ x \mid x = 2^n, n \text{ is a natural number and } n \leq 6 \} \)