Certainly! Let's go through each part step-by-step.
### Part (a)
- Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } x > -3 \}\)[/tex]
This set includes all integers that are greater than -3. Let's rewrite this in roster form by listing the first few elements.
- Roster form: [tex]\(\{0, 1, 2, 3, 4, \ldots\}\)[/tex]
The ellipsis ([tex]\(...\)[/tex]) indicates that this pattern continues indefinitely in a positive direction.
### Part (b)
- Roster form: [tex]\(\{-6, -5, -4, -3, -2, -1\}\)[/tex]
This set includes specific negative integers from -6 to -1. Now let's convert this into set-builder form.
- Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } -7 < x < 0\}\)[/tex]
This specifies that [tex]\(x\)[/tex] is an integer such that it is greater than -7 and less than 0, which captures all the elements listed in the roster form.
So, the final answers are:
(a)
- Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } x > -3 \}\)[/tex]
- Roster form: [tex]\(\{0, 1, 2, 3, 4, \ldots\}\)[/tex]
(b)
- Roster form: [tex]\(\{-6, -5, -4, -3, -2, -1\}\)[/tex]
- Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } -7 < x < 0\}\)[/tex]
