Interval (mathematics)
An interval in mathematics is defined as a set of real numbers that lies between two specified endpoints, which can either be finite or infinite. Typically represented in bracket notation, such as [a, b], intervals include all numbers from the smaller endpoint a to the larger endpoint b. Depending on whether the endpoints are included, intervals can be classified as closed (including endpoints) or open (excluding endpoints). There are also mixed types, such as left-bounded or right-bounded intervals, which specify whether one endpoint is finite or infinite. For example, intervals can be used to represent age ranges for school enrollment or price ranges for products based on market conditions. Special cases include degenerate intervals that consist of a single real number, and proper intervals, which contain an infinite number of elements. Overall, intervals play a crucial role in various mathematical contexts, including real analysis, and have practical applications in everyday scenarios.
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Interval (mathematics)
An interval is a set of real numbers, defined as those numbers between (and by default, including) two designated endpoints—though one of the endpoints may be infinite, which means that the real numbers themselves, taken as a set, are an interval, as is the set of all positive real numbers. The empty set is also an interval. Typically, though, an interval will be given as the numbers between a and b.
The conventional representation of an interval is [a, b], where a and b are the endpoints and a is usually the smaller endpoint. It is usually assumed that the interval includes the endpoints and the numbers between them: that [a, b] is the set of numbers greater than or equal to a and less than or equal to b. To exclude an endpoint, the bracket is usually reversed: ]a, b]; [a, b[; or ]a, b[, for instance. In some contexts, parentheses are used instead of brackets to indicate intervals excluding their endpoints. In some publications—because some countries use a decimal comma in numeric notation—a semicolon is used instead of a comma to separate the endpoints of the interval. In some cases when an interval consists only of the integers in the set (especially common in computer programming), the notation [a .. b] is used instead of a comma.
Overview
Intervals are frequently encountered in everyday life. Children entering first grade are often limited to those who turn 6 between November 1 (for instance) of one year and October 31 of the next, essentially an interval consisting of a set of birthdays. Public streets permit driving within an interval bound by the speed limit at one endpoint and an unofficial minimum speed at the other endpoint. The possible price of a particular product in a particular area is an interval, the endpoints of which are determined by various economic factors: For instance, the maximum price (one endpoint) of a book is its cover price, but it may be put on sale by the bookstore for as little as 50% of that price (the other endpoint) or may be found used for even less. And gasoline varies by price even on the same city block, as gas stations compete with one another for customers.
Intervals that exclude the endpoints are often called open intervals, while endpoint-inclusive intervals are closed. If its smallest endpoint is not negative infinity, it is left-bounded; if its largest endpoint is not infinity, it is right-bounded; by extension, if it is both left- and right-bounded, both of its endpoints are finite real numbers—and it is called a bounded interval or finite interval. An interval consisting of only a single real number is a degenerate interval, which some mathematicians consider to include the empty set. A real interval that is neither degenerate nor the empty set is called a proper interval, and with the exception of the computer programming cases mentioned above, consists of an infinite number of elements.
Bibliography
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Hanna, Gila. Explanation and Proof in Mathematics. New York: Springer, 2014.
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Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.