Let's go through each part of the problem step-by-step.
### Part (a)
We are given a set in roster form, which lists out elements of the set explicitly:
Given Roster Form:
[tex]\[ \{0, 1, 2, 3, \ldots\} \][/tex]
This set includes all non-negative integers starting from 0. To convert this to set-builder form, we need to describe this set using a property that all the elements of the set satisfy.
Set-builder Form:
[tex]\[ \{x \mid x \text{ is an integer and } x \geq 0\} \][/tex]
Explanation: The set-builder notation describes the set containing all integers [tex]\(x\)[/tex] that are greater than or equal to 0.
### Part (b)
We are given a set in set-builder form, which uses a property to define the set:
Given Set-builder Form:
[tex]\[ \{y \mid y \text{ is an integer and } -3 \leq y \leq 0\} \][/tex]
This set includes all integers [tex]\(y\)[/tex] that satisfy the condition [tex]\(-3 \leq y \leq 0\)[/tex]. To convert this to roster form, we need to list out all the elements that satisfy this condition.
Roster Form:
[tex]\[\{-3, -2, -1, 0\}\][/tex]
Explanation: The roster form shows explicitly that the set includes the integers -3, -2, -1, and 0.
### Summary
- Set given in roster form: [tex]\(\{0, 1, 2, 3, \ldots\}\)[/tex]
- Converted to set-builder form: [tex]\(\{x \mid x \text{ is an integer and } x \geq 0\}\)[/tex]
- Set given in set-builder form: [tex]\(\{y \mid y \text{ is an integer and } -3 \leq y \leq 0\}\)[/tex]
- Converted to roster form: [tex]\(\{-3, -2, -1, 0\}\)[/tex]
