To find the correct set-builder notation for the given set [tex]\(\{0, 1, 2, 3, 4, 5, \ldots\}\)[/tex], let's examine the properties and elements of this set in detail.
1. Identifying the Elements:
- The given set includes 0, 1, 2, 3, 4, 5, and so on, which are all non-negative integers (starting from 0 and increasing without bound).
2. Verifying each choice:
- Choice 1: [tex]\(\{x: x \text{ is an integer and } x \geq 0\}\)[/tex]
- This notation represents all integers [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is greater than or equal to 0. This matches the given set perfectly as it starts from 0 and includes all increasing integers.
- Choice 2: [tex]\(\{x: x \text{ is an integer and } -0 \leq x \leq 5\}\)[/tex]
- This notation represents all integers [tex]\(x\)[/tex] within the range [tex]\(-0 \leq x \leq 5\)[/tex]. While [tex]\(-0\)[/tex] is just 0, the range only includes integers from 0 to 5. This does not match our given set which continues infinitely beyond 5.
- Choice 3: [tex]\(\{x: x \text{ is an integer and } x < 0\}\)[/tex]
- This notation represents all integers [tex]\(x\)[/tex] that are less than 0. This is the opposite of our given set, which only includes non-negative integers.
- Choice 4: [tex]\(\{x: x \text{ is an integer and } x \leq 0\}\)[/tex]
- This notation represents all integers [tex]\(x\)[/tex] less than or equal to 0. Our set includes integers starting from 0 and increasing, not those less than 0.
3. Conclusion:
- After reviewing each option, the correct set-builder form for the set [tex]\(\{0, 1, 2, 3, 4, 5, \ldots\}\)[/tex] is clearly:
[tex]\(\{x: x \text{ is an integer and } x \geq 0\}\)[/tex]
Thus, the correct choice is the first option, and the index of this correct choice is 1.
