Write each of the following sets in Roster form and write the cardinal number of each .
` J= { x: x = (1)/( n) , n in N, 4 lt n lt 10 } `

Answer

To solve the problem, we need to write the set \( J \) in roster form and find its cardinal number. Let's break down the steps:

### Step 1: Identify the values of \( n \)
The set \( J \) is defined as:
\[ J = \{ x : x = \frac{1}{n}, n \in \mathbb{N}, 4 < n < 10 \} \]

Here, \( n \) is a natural number (denoted by \( \mathbb{N} \)) that is greater than 4 and less than 10. The natural numbers that satisfy this condition are:
\[ n = 5, 6, 7, 8, 9 \]

### Step 2: Calculate the corresponding values of \( x \)
Now, we will substitute each value of \( n \) into the equation \( x = \frac{1}{n} \):
- For \( n = 5 \): \( x = \frac{1}{5} \)
- For \( n = 6 \): \( x = \frac{1}{6} \)
- For \( n = 7 \): \( x = \frac{1}{7} \)
- For \( n = 8 \): \( x = \frac{1}{8} \)
- For \( n = 9 \): \( x = \frac{1}{9} \)

### Step 3: Write the set in roster form
Now that we have the values of \( x \), we can write the set \( J \) in roster form:
\[ J = \left\{ \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9} \right\} \]

### Step 4: Determine the cardinal number
The cardinal number of a set is the number of elements in that set. In this case, we have the following elements in set \( J \):
- \( \frac{1}{5} \)
- \( \frac{1}{6} \)
- \( \frac{1}{7} \)
- \( \frac{1}{8} \)
- \( \frac{1}{9} \)

Thus, the cardinal number of set \( J \) is:
\[ |J| = 5 \]

### Final Answer
The set \( J \) in roster form is:
\[ J = \left\{ \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9} \right\} \]
The cardinal number of set \( J \) is:
\[ |J| = 5 \]

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