Express the following sets in set-builder form :
(i) `(7/8, 8/9, 9/10, 10/11, 11/12)`,
(ii) {0,3,56,9,12,15,18}
(iii) `{1/3, 1/9, 1/17, 1/81, 1/243}`
(iv) `[x: x^(2) -6x-7=0]`

Answer

To express the given sets in set-builder form, we will analyze each set and identify a general rule or function that describes the elements of the set.

### Step-by-Step Solution:

**(i)** For the set `(7/8, 8/9, 9/10, 10/11, 11/12)`

1. **Identify the pattern:** The elements can be expressed as fractions where the numerator is one less than the denominator.
2. **Generalize the elements:** We can express the elements as \( x = \frac{n}{n+1} \) where \( n \) is a natural number starting from 7 to 11.
3. **Set-builder form:** Thus, the set can be written as:
\[ \{ x : x = \frac{n}{n+1}, n \in \mathbb{N}, 7 \leq n \leq 11 \} \]

**(ii)** For the set `{0, 3, 6, 9, 12, 15, 18}`

1. **Identify the pattern:** The elements are multiples of 3.
2. **Generalize the elements:** We can express the elements as \( x = 3n \) where \( n \) is a whole number starting from 0 to 6.
3. **Set-builder form:** Thus, the set can be written as:
\[ \{ x : x = 3n, n \in \mathbb{W}, 0 \leq n \leq 6 \} \]

**(iii)** For the set `{1/3, 1/9, 1/27, 1/81, 1/243}`

1. **Identify the pattern:** The denominators are powers of 3.
2. **Generalize the elements:** We can express the elements as \( x = \frac{1}{3^n} \) where \( n \) is a natural number starting from 1 to 5.
3. **Set-builder form:** Thus, the set can be written as:
\[ \{ x : x = \frac{1}{3^n}, n \in \mathbb{N}, 1 \leq n \leq 5 \} \]

**(iv)** For the set `[x: x^2 - 6x - 7 = 0]`

1. **Identify the equation:** This is a quadratic equation.
2. **Find the roots:** We can factor the equation as \( (x - 7)(x + 1) = 0 \). Thus, the roots are \( x = 7 \) and \( x = -1 \).
3. **Set-builder form:** Thus, the set can be written as:
\[ \{ x : x = -1 \text{ or } x = 7 \} \]

### Final Set-Builder Forms:

1. **(i)** \(\{ x : x = \frac{n}{n+1}, n \in \mathbb{N}, 7 \leq n \leq 11 \}\)
2. **(ii)** \(\{ x : x = 3n, n \in \mathbb{W}, 0 \leq n \leq 6 \}\)
3. **(iii)** \(\{ x : x = \frac{1}{3^n}, n \in \mathbb{N}, 1 \leq n \leq 5 \}\)
4. **(iv)** \(\{ x : x = -1 \text{ or } x = 7 \}\)