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Write each set in the indicated form. If you need to use "..." to indicate a pattern, make sure to list at least four elements of the set.

(a) Roster form: [tex]\{-1, 0, 1, 2, 3\}[/tex]
Set-builder form: [tex]\{x \mid x \in \mathbb{Z}, -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, \ldots\}[/tex]


Sagot :

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]