Write the following intervals in set builder form :
(i) `(-3,2)`
(ii) `(-3,5]`
(iii) `[1,3)`
(iv) `[2,12]`

Answer

To express the given intervals in set builder form, we will follow a systematic approach for each interval.

### Solution:

**(i) For the interval `(-3, 2)`**

1. Identify the interval: The interval is open on both ends, meaning it does not include -3 and 2.
2. Set builder form: We express this as:
\[ \{ x \in \mathbb{R} \mid -3 < x < 2 \} \]

**(ii) For the interval `(-3, 5]`**

1. Identify the interval: The interval is open on the left at -3 and closed on the right at 5, meaning it does not include -3 but includes 5.
2. Set builder form: We express this as:
\[ \{ x \in \mathbb{R} \mid -3 < x \leq 5 \} \]

**(iii) For the interval `[1, 3)`**

1. Identify the interval: The interval is closed on the left at 1 and open on the right at 3, meaning it includes 1 but does not include 3.
2. Set builder form: We express this as:
\[ \{ x \in \mathbb{R} \mid 1 \leq x < 3 \} \]

**(iv) For the interval `[2, 12]`**

1. Identify the interval: The interval is closed on both ends, meaning it includes both 2 and 12.
2. Set builder form: We express this as:
\[ \{ x \in \mathbb{R} \mid 2 \leq x \leq 12 \} \]

### Final Set Builder Forms:
1. **(i)** \(\{ x \in \mathbb{R} \mid -3 < x < 2 \}\)
2. **(ii)** \(\{ x \in \mathbb{R} \mid -3 < x \leq 5 \}\)
3. **(iii)** \(\{ x \in \mathbb{R} \mid 1 \leq x < 3 \}\)
4. **(iv)** \(\{ x \in \mathbb{R} \mid 2 \leq x \leq 12 \}\)