Interval

Interval

Set of numbers included between two given numbers called the endpoints of the interval.

The length of an interval is the absolute value of the difference between the upper endpoint and the lower endpoint of an interval. It is also called the amplitude of the interval.

Symbols and graphs

Intervals may be written in various ways, depending on the characteristics of their endpoints a and b:

Consider a continuous set E of real numbers.

• An open interval of E with endpoints a and b is a subset of E written as (a, b) and consisting of all the elements of E strictly included between a and b. It is formally written as:
(a, b) = {x ∈ E | $$a$$ < x < $$b$$}.
An open interval is graphically represented by a segment whose endpoints are open circles.

• A left half-open interval of E with endpoints a and b is a subset of E written as (a, b] and containing all the elements of E strictly greater than a and less than or equal to b. It is formally written as:
(a, b] = {x ∈ E | $$a$$ < x ≤ $$b$$}.
A left half-open interval is graphically represented by a segment whose left endpoint is open and whose right endpoint is closed.

• A right half-open interval of E with endpoints a and b is a subset of E written as [a, b) and containing all the elements of E greater than or equal to a and strictly less than b. It is formally written as:
[a, b) = {x ∈ E | $$a$$ ≤ x < $$b$$}.
A right half-open interval is graphically represented by a segment whose left endpoint is closed and whose right endpoint is open.

• A closed interval [A,B] of E is an interval whose endpoints are included. A closed interval is represented by a segment whose two endpoints are closed points. A closed interval is graphically represented by a segment whose two endpoints are closed.

Examples

• Consider the interval [12, 35]. The length of this interval is 23, since |35 – 12| = 23.
• Consider the interval [–12, –35]. The length of this interval is 23, since |–35 – (–12)| = 23.
• Consider the interval [–12, 35]. The length of this interval is 47, since |35 – (–12)| = 47.