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Concept Version 12
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Interval Notation

Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.

Learning Objective

  • Use interval notation to show how a set of numbers is bounded


Key Points

    • A real interval is a set of real numbers with the property that any number that lies between two numbers included in the set is also included in the set.
    • The interval of numbers between $a$ and $b$, including $a$ and $b$, is denoted $[a,b]$. The two numbers $a$ and $b$ are called the endpoints of the interval.
    • To indicate that an endpoint of a set is not included in the set, the square bracket enclosing the endpoint can be replaced with a parenthesis.
    • An open interval does not include its endpoints, and is enclosed in parentheses. A closed interval includes its endpoints, and is enclosed in square brackets.
    • An interval is considered bounded if both endpoints are real numbers. An interval is unbounded if both endpoints are not real numbers.
    • Replacing an endpoint with positive or negative infinity—e.g., $(- \infty, b]$—indicates that a set is unbounded in one direction, or half-bounded.

Terms

  • unbounded interval

    A set for which neither endpoint is a real number.

  • open interval

    A set of real numbers that does not include its endpoints.

  • closed interval

    A set of real numbers that includes both of its endpoints.

  • half-bounded interval

    A set for which one endpoint is a real number and the other is not.

  • bounded interval

    A set for which both endpoints are real numbers.

  • interval

    A distance in space.

  • endpoint

    Either of the two points at the ends of a line segment.


Full Text

A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers $x$ satisfying $0 \leq x \leq 1$ is an interval that contains 0 and 1, as well as all the numbers between them. Other examples of intervals include the set of all real numbers and the set of all negative real numbers.

The interval of numbers between $a$ and $b$, including $a$ and $b$, is often denoted $[a,b]$. The two numbers are called the endpoints of the interval. 

Open and Closed Intervals

An open interval does not include its endpoints and is indicated with parentheses. For example, $(0,1)$ describes an interval greater than 0 and less than 1. 

A closed interval includes its endpoints and is denoted with square brackets rather than parentheses. For example, $[0,1]$ describes an interval greater than or equal to 0 and less than or equal to 1.

To indicate that only one endpoint of an interval is included in that set, both symbols will be used. For example, the interval of numbers between 1 and 5, including 1 but excluding 5, is written as $[1,5)$.

The image below illustrates open and closed intervals on a number line.

Intervals

Representations of open and closed intervals on the real number line.

Bounded and Unbounded Intervals

An interval is said to be bounded if both of its endpoints are real numbers. Bounded intervals are also commonly known as finite intervals. Conversely, if neither endpoint is a real number, the interval is said to be unbounded. For example, the interval $(1,10)$ is considered bounded; the interval $(- \infty, + \infty)$ is considered unbounded.

The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded.

An interval that has only one real-number endpoint is said to be half-bounded, or more descriptively, left-bounded or right-bounded. For example, the interval $(1, + \infty)$ is half-bounded; specifically, it is left-bounded.

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