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.
