Mathematical Intervals
an interval is a set of numbers between two given numbers, often representing a portion of the real number line. Intervals are commonly used in algebra, calculus, and real analysis.
1. Open Interval (a, b)
An open interval (a, b) includes all numbers between a and b, but **not** a and b themselves.
Example: (2, 5) means all numbers between 2 and 5, but **not** 2 and 5.
2. Closed Interval [a, b]
A closed interval [a, b] includes all numbers between a and b, **including** a and b.
Example: [2, 5] includes 2, 3, 4, and 5.
3. Half-Open (or Half-Closed) Interval
A half-open interval includes one endpoint but not the other:
- [a, b) → Includes **a**, but **not** b.
- (a, b] → Includes **b**, but **not** a.
Example: [2, 5) includes 2 but not 5.
4. Infinite Intervals
When an interval extends indefinitely:
- (a, ∞) → All values greater than a.
- [a, ∞) → All values greater than or equal to a.
- (-∞, b) → All values less than b.
- (-∞, b] → All values less than or equal to b.
Example: (3, ∞) means all numbers greater than 3.
Graphical Representation of Intervals
1. Open Interval (a, b)
An open interval (a, b) includes all numbers between a and b, but **not** a and b.
2. Closed Interval [a, b]
A closed interval [a, b] includes all numbers between a and b, **including** a and b.
3. Half-Open Interval [a, b) and (a, b]
One endpoint is included, and the other is not.
4. Infinite Interval (a, ∞) and (-∞, b]
When an interval extends indefinitely.
