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 these steps:

### Solution Steps:

1. **For the interval \((-3, 2)\)**:
- This interval includes all real numbers \(x\) such that \(x\) is greater than \(-3\) and less than \(2\).
- In set builder notation, this can be expressed as:
\[ \{ x \in \mathbb{R} \mid -3 < x < 2 \} \]

2. **For the interval \((-3, 5]\)**:
- This interval includes all real numbers \(x\) such that \(x\) is greater than \(-3\) and less than or equal to \(5\).
- In set builder notation, this can be expressed as:
\[ \{ x \in \mathbb{R} \mid -3 < x \leq 5 \} \]

3. **For the interval \([1, 3)\)**:
- This interval includes all real numbers \(x\) such that \(x\) is greater than or equal to \(1\) and less than \(3\).
- In set builder notation, this can be expressed as:
\[ \{ x \in \mathbb{R} \mid 1 \leq x < 3 \} \]

4. **For the interval \([2, 12]\)**:
- This interval includes all real numbers \(x\) such that \(x\) is greater than or equal to \(2\) and less than or equal to \(12\).
- In set builder notation, this can be expressed as:
\[ \{ x \in \mathbb{R} \mid 2 \leq x \leq 12 \} \]

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