Certainly! Let's express the given sets in set-builder notation step-by-step.
### (i) [tex]\(\{4,8,12,16,20\}\)[/tex]
We need to observe a pattern in the set's elements:
- The set is [tex]\(\{4, 8, 12, 16, 20\}\)[/tex].
- We can see that each element is a multiple of 4.
- If we let [tex]\(n\)[/tex] be a natural number (elements of [tex]\(\mathbb{N}\)[/tex]), such that [tex]\(1 \leq n \leq 5\)[/tex], the numbers 4, 8, 12, 16, and 20 can be written as [tex]\(4 \times n\)[/tex].
Therefore, the set can be expressed in set-builder notation as:
[tex]\[ \{x \mid x = 4n, \, n \in \mathbb{N}, \, 1 \leq n \leq 5\} \][/tex]
So, the set-builder form is:
[tex]\[ \text{Set A} = \{x \mid x = 4n, \, n \in \mathbb{N}, \, 1 \leq n \leq 5\} \][/tex]
### (ii) [tex]\(\{-15\}\)[/tex]
This set contains only one element, which is [tex]\(-15\)[/tex].
- We simply state that [tex]\(x\)[/tex] is exactly [tex]\(-15\)[/tex].
Therefore, the set can be expressed in set-builder notation as:
[tex]\[ \{x \mid x = -15\} \][/tex]
So, the set-builder form is:
[tex]\[ \text{Set B} = \{x \mid x = -15\} \][/tex]
### (iii) [tex]\(\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\}\)[/tex]
We need to observe a pattern in the set's elements:
- The set is [tex]\(\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\right\}\)[/tex].
- Each element can be represented as [tex]\(\frac{n}{n-1}\)[/tex], where [tex]\(n\)[/tex] starts from 2 and increases in the set of natural numbers.
Therefore, the set can be expressed in set-builder notation as:
[tex]\[ \{x \mid x = \frac{n}{n-1}, \, n \in \mathbb{N}, \, n \geq 2\} \][/tex]
So, the set-builder form is:
[tex]\[ \text{Set C} = \{x \mid x = \frac{n}{n-1}, \, n \in \mathbb{N}, \, n \geq 2\} \][/tex]
In conclusion, the sets in set-builder notation are:
1. [tex]\(\text{Set A} = \{x \mid x = 4n, \, n \in \mathbb{N}, \, 1 \leq n \leq 5\}\)[/tex]
2. [tex]\(\text{Set B} = \{x \mid x = -15\}\)[/tex]
3. [tex]\(\text{Set C} = \{x \mid x = \frac{n}{n-1}, \, n \in \mathbb{N}, \, n \geq 2\}\)[/tex]
