Write each of the following sets in set- builder form :
` E= {(1)/(3),(1)/(5) , (1)/(7) ,( 1)/(11),( 1)/(13),(1)/(17),(1)/(19) } `

Answer

To express the set \( E = \left\{ \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{17}, \frac{1}{19} \right\} \) in set-builder form, we need to identify a common property that all the elements share.

### Step-by-Step Solution:

1. **Identify the Elements**: The elements of the set \( E \) are \( \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{17}, \frac{1}{19} \).

2. **Observe the Denominators**: The denominators of these fractions are \( 3, 5, 7, 11, 13, 17, \) and \( 19 \).

3. **Recognize the Pattern**: Notice that all these denominators are prime numbers.

4. **Define the Set in Set-Builder Notation**: We can express the set in set-builder form by stating that \( E \) consists of elements \( x \) such that \( x \) can be represented as \( \frac{1}{n} \) where \( n \) is a prime number.

5. **Specify the Range for \( n \)**: Since all the prime numbers in the denominators are between 2 and 20, we can specify that \( n \) should be a prime number that lies between 2 and 20.

6. **Final Set-Builder Form**: Thus, we can write the set \( E \) in set-builder notation as:
\[ E = \{ x \mid x = \frac{1}{n}, \, n \text{ is a prime number and } 2 < n < 20 \} \]