Experience the convenience of getting your questions answered at IDNLearn.com. Join our knowledgeable community to find the answers you need for any topic or issue.

3. Express the following sets in set-builder notation:

(i) [tex]\{4, 8, 12, 16, 20\}[/tex]

(ii) [tex]\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\}[/tex]


Sagot :

Sure! Let's express the given sets in set-builder notation.

### (i) [tex]\(\{4,8,12,16,20\}\)[/tex]

To express the set [tex]\(\{4,8,12,16,20\}\)[/tex] using set-builder notation, we notice that each element in the set is a multiple of 4. Specifically, the elements are of the form [tex]\(4n\)[/tex] where [tex]\(n\)[/tex] is an integer. We also observe that [tex]\(n\)[/tex] runs from 1 to 5 inclusive.

Thus, in set-builder notation, we can write:

[tex]\[ \{4,8,12,16,20\} = \{ x \mid x = 4n, n \in \mathbb{Z}, 1 \leq n \leq 5 \} \][/tex]

Where:
- [tex]\(x\)[/tex] represents the elements of the set.
- [tex]\(4n\)[/tex] defines the structure of each element in the set.
- [tex]\(n \in \mathbb{Z}\)[/tex] means [tex]\(n\)[/tex] is an integer.
- [tex]\(1 \leq n \leq 5\)[/tex] limits [tex]\(n\)[/tex] to the values 1, 2, 3, 4, and 5.

### (iii) [tex]\(\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\}\)[/tex]

To express the set [tex]\(\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\}\)[/tex] using set-builder notation, we observe a pattern where each entry [tex]\(x\)[/tex] in the sequence is in the form [tex]\(\frac{n}{n-1}\)[/tex] where [tex]\(n\)[/tex] is an integer starting from 2.

Thus, in set-builder notation, we can write:

[tex]\[ \left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\} = \{ x \mid x = \frac{n}{n-1}, n \in \mathbb{Z}, n \geq 2 \} \][/tex]

Where:
- [tex]\(x\)[/tex] represents the elements of the set.
- [tex]\(\frac{n}{n-1}\)[/tex] defines the structure of each element in the set.
- [tex]\(n \in \mathbb{Z}\)[/tex] means [tex]\(n\)[/tex] is an integer.
- [tex]\(n \geq 2\)[/tex] limits [tex]\(n\)[/tex] to integers starting from 2.

Both sets are now expressed in set-builder notation based on the observed patterns.