State whether each of the following sets is a finite set or an infinite set:
`{x : x=3n-2, n in W, n le 8}`

Answer

To determine whether the set `{x : x = 3n - 2, n in W, n ≤ 8}` is finite or infinite, we will follow these steps:

### Step 1: Understand the definition of the set
The set is defined as `{x : x = 3n - 2, n in W, n ≤ 8}`. Here, `W` represents the set of whole numbers, which includes 0, 1, 2, 3, and so on.

### Step 2: Identify the values of `n`
Since `n` is a whole number and must be less than or equal to 8, the possible values for `n` are:
- 0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8

### Step 3: Calculate the corresponding values of `x`
Now, we will calculate the value of `x` for each value of `n`:

- For `n = 0`:
\( x = 3(0) - 2 = -2 \)

- For `n = 1`:
\( x = 3(1) - 2 = 1 \)

- For `n = 2`:
\( x = 3(2) - 2 = 4 \)

- For `n = 3`:
\( x = 3(3) - 2 = 7 \)

- For `n = 4`:
\( x = 3(4) - 2 = 10 \)

- For `n = 5`:
\( x = 3(5) - 2 = 13 \)

- For `n = 6`:
\( x = 3(6) - 2 = 16 \)

- For `n = 7`:
\( x = 3(7) - 2 = 19 \)

- For `n = 8`:
\( x = 3(8) - 2 = 22 \)

### Step 4: List the values of `x`
The values of `x` we calculated are:
- -2
- 1
- 4
- 7
- 10
- 13
- 16
- 19
- 22

### Step 5: Count the number of elements
Now, we count the distinct values of `x`:
- The set is: {-2, 1, 4, 7, 10, 13, 16, 19, 22}

There are a total of 9 distinct elements in this set.

### Step 6: Determine if the set is finite or infinite
Since we have counted a finite number of elements (9 elements), we can conclude that the set is a finite set.

### Final Answer
The set `{x : x = 3n - 2, n in W, n ≤ 8}` is a finite set.

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