Write each of the following sets in set- builder form :
` D = {(1)/(2),(2)/(3),(3)/(4) ,….(8)/(9) }.`

Answer

To express the set \( D = \left\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots, \frac{8}{9} \right\} \) in set-builder form, we follow these steps:

### Step-by-Step Solution:

1. **Identify the Pattern**:
The elements of the set \( D \) are fractions where the numerator is a natural number starting from 1 and ending at 8, and the denominator is one more than the numerator.

2. **Define the Variable**:
Let \( n \) be a natural number that represents the numerator of the fractions.

3. **Write the Expression for the Elements**:
Each element of the set can be represented as \( \frac{n}{n+1} \).

4. **Specify the Domain of \( n \)**:
Since \( n \) starts from 1 and goes up to 8 (inclusive), we can write \( n \) such that \( n \in \mathbb{N} \) and \( 1 \leq n \leq 8 \).

5. **Combine into Set-Builder Notation**:
Now we can express the set \( D \) in set-builder notation as:
\[ D = \{ x \mid x = \frac{n}{n+1}, n \in \mathbb{N}, 1 \leq n \leq 8 \} \]

### Final Answer:
Thus, the set \( D \) in set-builder form is:
\[ D = \{ x \mid x = \frac{n}{n+1}, n \in \mathbb{N}, 1 \leq n \leq 8 \} \]