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Use the listing method to represent the following set.

[tex]\[
\{x \mid x \in \mathbb{N}, x \leq 3\}
\][/tex]

A. [tex]\(\{3,4,5,6, \ldots\}\)[/tex]

B. [tex]\(\{\ldots, 0,1,2,3\}\)[/tex]

C. [tex]\(\{4,5,6,7, \ldots\}\)[/tex]


Sagot :

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.