Sure! Let's use the listing method to represent the given sets one by one.
### Set 1: [tex]\(\{x \mid x \in 1, x \leq 3\}\)[/tex]
This set notation indicates that [tex]\(x\)[/tex] is a positive integer and [tex]\(x \leq 3\)[/tex]. Listing all such integers, we have:
[tex]\[
\{1, 2, 3\}
\][/tex]
### Set 2: [tex]\(\{3, 4, 5, 6, \ldots\}\)[/tex]
This set includes all integers greater than or equal to 3. Listing the first few elements:
[tex]\[
\{3, 4, 5, 6, 7\}
\][/tex]
Note that the listing shows the beginning of the sequence, but the set continues indefinitely beyond 7.
### Set 3: [tex]\(\{\ldots, 0, 1, 2, 3\}\)[/tex]
This set includes all integers less than or equal to 3. Listing a few elements, both large negative and small non-negative ones, we have:
[tex]\[
\{-3, -2, -1, 0, 1, 2, 3\}
\][/tex]
Again, note that the listing provides some representative elements and the set continues indefinitely towards negative values.
### Set 4: [tex]\(\{4, 5, 6, 7, \ldots\}\)[/tex]
This set includes all integers greater than or equal to 4. Listing the first few elements:
[tex]\[
\{4, 5, 6, 7\}
\][/tex]
As with Set 2, this listing indicates the beginning of an infinite sequence.
In summary:
- The first set is [tex]\(\{1, 2, 3\}\)[/tex].
- The second set is [tex]\(\{3, 4, 5, 6, 7\}\)[/tex].
- The third set is [tex]\(\{-3, -2, -1, 0, 1, 2, 3\}\)[/tex].
- The fourth set is [tex]\(\{4, 5, 6, 7\}\)[/tex].
These listings represent the sets according to the given specifications.
