To convert the given intervals into set-builder form, we will follow the properties of the intervals, specifically noting whether endpoints are included or excluded based on the type of brackets used.
### Step-by-Step Solution:
**(i)** For the interval \( A = (-2, 3) \):
- Since the interval uses round brackets, the endpoints -2 and 3 are not included.
- Set-builder form:
\[
A = \{ x \mid x \in \mathbb{R}, -2 < x < 3 \}
\]
**(ii)** For the interval \( B = [4, 10] \):
- This interval uses square brackets, meaning both endpoints 4 and 10 are included.
- Set-builder form:
\[
B = \{ x \mid x \in \mathbb{R}, 4 \leq x \leq 10 \}
\]
**(iii)** For the interval \( C = [-1, 8] \):
- Again, square brackets indicate that both endpoints -1 and 8 are included.
- Set-builder form:
\[
C = \{ x \mid x \in \mathbb{R}, -1 \leq x \leq 8 \}
\]
**(iv)** For the interval \( D = (4, 9] \):
- The round bracket indicates that 4 is not included, while the square bracket indicates that 9 is included.
- Set-builder form:
\[
D = \{ x \mid x \in \mathbb{R}, 4 < x \leq 9 \}
\]
**(v)** For the interval \( E = [-10, 0) \):
- Here, -10 is included (square bracket) and 0 is not included (round bracket).
- Set-builder form:
\[
E = \{ x \mid x \in \mathbb{R}, -10 \leq x < 0 \}
\]
**(vi)** For the interval \( F = (0, 5] \):
- The round bracket indicates that 0 is not included, while the square bracket indicates that 5 is included.
- Set-builder form:
\[
F = \{ x \mid x \in \mathbb{R}, 0 < x \leq 5 \}
\]
### Summary of Set-Builder Forms:
1. \( A = \{ x \mid x \in \mathbb{R}, -2 < x < 3 \} \)
2. \( B = \{ x \mid x \in \mathbb{R}, 4 \leq x \leq 10 \} \)
3. \( C = \{ x \mid x \in \mathbb{R}, -1 \leq x \leq 8 \} \)
4. \( D = \{ x \mid x \in \mathbb{R}, 4 < x \leq 9 \} \)
5. \( E = \{ x \mid x \in \mathbb{R}, -10 \leq x < 0 \} \)
6. \( F = \{ x \mid x \in \mathbb{R}, 0 < x \leq 5 \} \)
