Understanding Intervals and Interval Notation

Key Concepts

Interval: A range of numbers on the number line.
Endpoints: The starting and ending points of an interval.
Closed Interval: Includes the endpoints.
- Denoted by filled-in circles on a number line.
- Written with brackets: [a, b].
- Example: The interval from -3 to 2 including -3 and 2 is shown as [-3, 2].
Open Interval: Does not include the endpoints.
- Denoted by open circles on a number line.
- Written with parentheses: (a, b).
- Example: The interval from -1 to 4 excluding -1 and 4 is shown as (-1, 4).

Half-Open Interval: Includes only one of the endpoints.
- Example: If -4 is excluded and -1 is included, it is denoted by (-4, -1].

Set Builder Notation: Uses curly brackets and a logical condition.
- Example: {x ∈ ℝ | -3 ≤ x ≤ 2} indicates a closed interval of all x between -3 and 2.
- Example: {x ∈ ℝ | -1 < x < 4} indicates an open interval between -1 and 4.
Mathematical Symbolism: Uses symbols like epsilon (∈) to denote membership in a set.

Excluding a Specific Point: Denote all real numbers except one.
- Example: {x ∈ ℝ | x ≠ 1} or {x ∈ ℝ | x < 1 or x > 1}.
Infinite Intervals: Involving infinity always uses parentheses.
- Example: (-∞, 1) ∪ (1, ∞) represents all real numbers except 1.

Infinity Notation: Always uses parentheses because infinity is not an actual endpoint.
Real Numbers: Typically denoted by ℝ and encompass all numbers on the number line.

By understanding these notations and concepts, you can accurately depict and interpret intervals in mathematical contexts.