Explain the defined intervals we have in real number system

Solution:
1. In the real number system, there are different ways to define intervals:
2. **Open Interval**:
- Notation: (a, b)
- This interval includes all real numbers x such that a < x < b.
- The endpoints a and b are not included.

3. **Closed Interval**:
- Notation: [a, b]
- This interval includes all real numbers x such that a \leq x \leq b.
- The endpoints a and b are included.

4. **Half-Open Interval (Left Open, Right Closed)**:
- Notation: (a, b]
- This interval includes all real numbers x such that a < x \leq b.
- The endpoint a is not included, but b is.

5. **Half-Open Interval (Left Closed, Right Open)**:
- Notation: [a, b)
- This interval includes all real numbers x such that a \leq x < b.
- The endpoint a is included, but b is not.

6. **Infinite Intervals**:
- **Interval from a point to infinity**: [a, \infty) or (a, \infty) where all numbers starting from (and possibly including) a to infinity are included.
- **Interval from negative infinity to a point**: (-\infty, b] or (-\infty, b) where all numbers up to (and possibly including) b are included.

7. Each interval type has a distinct way of indicating which endpoints are excluded or included.