Write the set `{1/2,2/3,3/4,4/5,5/6,6/7,7/8,8/9,9/(10)}` in the set builder form.

Options

A. `{x:x=n/(n+1),where AA n in N and 1le n le 9}`
B. `{x:x=n/(n-1),where AA n in N and 1le n le 9}`
C. `{x:x=n/(n+1),where AA n in N and 1lt n lt 9}`
D. none of these

Explanation

To express the set \(\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10} \}\) in set builder form, we follow these steps:

### Step 1: Identify the Pattern
First, we observe the elements in the set. Each element can be expressed as a fraction where the numerator is a natural number \(n\) and the denominator is \(n + 1\).

### Step 2: Define the General Element
We can denote the general element of the set as:
\[ x = \frac{n}{n + 1} \]
where \(n\) is a natural number.

### Step 3: Determine the Range of \(n\)
Next, we need to determine the appropriate values for \(n\) that will give us all the elements in the set. Observing the set, we see that:
- The smallest value of \(n\) that gives us the first element \(\frac{1}{2}\) is \(n = 1\).
- The largest value of \(n\) that gives us the last element \(\frac{9}{10}\) is \(n = 9\).

Thus, \(n\) can take values from \(1\) to \(9\).

### Step 4: Write the Set Builder Notation
Now we can write the set in set builder notation. The set can be expressed as:
\[ S = \{ x \mid x = \frac{n}{n + 1}, n \in \mathbb{N}, 1 \leq n \leq 9 \} \]

### Final Answer
Thus, the set \(\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10} \}\) in set builder form is:
\[ S = \{ x \mid x = \frac{n}{n + 1}, n \in \mathbb{N}, 1 \leq n \leq 9 \} \]