Let's tackle the question step by step.
### Part (a)
#### Given:
Roster form: [tex]\(\{-1,0,1,2,3\}\)[/tex]
#### Required:
Set-builder form.
To express this set in set-builder form, let's analyze its elements. The set contains the integers [tex]\(-1, 0, 1, 2,\)[/tex] and [tex]\(3\)[/tex].
From this, we can observe that it contains all the integers [tex]\(x\)[/tex] from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex], inclusive. Therefore, the set-builder notation can be written as:
[tex]\[
\{ x \mid x \text{ is an integer and } -1 \leq x \leq 3 \}
\][/tex]
### Part (b)
#### Given:
Set-builder form: [tex]\(\{ y \mid y \text{ is an integer and } y \geq -1 \}\)[/tex]
#### Required:
Roster form.
To convert from set-builder form to roster form, we'll list at least the first four elements to illustrate the pattern. Starting from [tex]\(-1\)[/tex], the next integers would be [tex]\(0\)[/tex], [tex]\(1\)[/tex], [tex]\(2\)[/tex], and so on.
So, the roster form representing the set including at least four elements to show the pattern would be:
[tex]\[
\{ -1, 0, 1, 2, 3, \ldots \}
\][/tex]
This indicates that [tex]\(y\)[/tex] continues indefinitely in the positive direction, starting from [tex]\(-1\)[/tex].
### Final Answer:
(a) Roster form: [tex]\(\{-1,0,1,2,3\}\)[/tex]
Set-builder form: [tex]\(\{ x \mid x \text{ is an integer and } -1 \leq x \leq 3 \}\)[/tex]
(b) Set-builder form: [tex]\(\{ y \mid y \text{ is an integer and } y \geq -1 \}\)[/tex]
Roster form: [tex]\(\{ -1, 0, 1, 2, 3, \ldots \}\)[/tex]
