Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. An interval can contain neither endpoint, either endpoint, or both endpoints.
For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].
Intervals are ubiquitous in
Intervals are likewise defined on an arbitrary
Definitions and terminology
An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset.
The endpoints of an interval are its
Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the least-upper-bound property of the real numbers. This characterization is used to specify intervals by mean of interval notation, which is described below.
An open interval does not include any endpoint, and is indicated with parentheses.[2] For example, is the interval of all real numbers greater than 0 and less than 1. (This interval can also be denoted by ]0, 1[, see below). The open interval (0, +∞) consists of real numbers greater than 0, i.e., positive real numbers. The open intervals are thus one of the forms
where and are real numbers such that When in the first case, the resulting interval is the empty set which is a degenerate interval (see below). The open intervals are those intervals that are open sets for the usual topology on the real numbers.
A closed interval is an interval that includes all its endpoints and is denoted with square brackets.[2] For example, [0, 1] means greater than or equal to 0 and less than or equal to 1. Closed intervals have one of the following forms in which a and b are real numbers such that
The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and are the only intervals that are both open and closed.
A half-open interval has two endpoints and includes only one of them. It is said left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals.[3] For example, (0, 1] means greater than 0 and less than or equal to 1, while [0, 1) means greater than or equal to 0 and less than 1. The half-open intervals have the form
Every closed interval is a
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval.[4][5]
A degenerate interval is any
An interval is said to be left-bounded or right-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals.
Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined).
The centre (midpoint) of a bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a − b|/2. These concepts are undefined for empty or unbounded intervals.
An interval is said to be left-open if and only if it contains no
An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has a maximum or is right unbounded; it is simply closed if it is both left-closed and right closed. So, the closed intervals coincide with the closed sets in that topology.
The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints.
For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X, and does not properly contain any other interval that also contains X.
An interval I is a subinterval of interval J if I is a
However, there is conflicting terminology for the terms segment and interval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics[7] defines interval (without a qualifier) to exclude both endpoints (i.e., open interval) and segment to include both endpoints (i.e., closed interval), while Rudin's Principles of Mathematical Analysis[8] calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half-open), regardless of whether endpoints are included.
Notations for intervals
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. In countries where numbers are written with a
Including or excluding endpoints
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in
Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the singleton set {a}. When a > b, all four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a, b) is often used to denote an
Some authors such as Yves Tillé use ]a, b[ to denote the complement of the interval (a, b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.
Infinite endpoints
In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with −∞ and +∞.
In this interpretation, the notations [−∞, b] , (−∞, b] , [a, +∞] , and [a, +∞) are all meaningful and distinct. In particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals.
Even in the context of the ordinary reals, one may use an
Integer intervals
When a and b are
Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis notation.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations like [a .. b) or [a .. b[ are rarely used for integer intervals.[citation needed]
Properties
The intervals are precisely the connected subsets of It follows that the image of an interval by any continuous function from to is also an interval. This is one formulation of the intermediate value theorem.
The intervals are also the convex subsets of The interval enclosure of a subset is also the convex hull of
The closure of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every connected subset of a topological space is a connected subset.) In other words, we have[10]
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example
If is viewed as a
Any element x of an interval I defines a partition of I into three disjoint intervals I1, I2, I3: respectively, the elements of I that are less than x, the singleton and the elements that are greater than x. The parts I1 and I3 are both non-empty (and have non-empty interiors), if and only if x is in the interior of I. This is an interval version of the
Dyadic intervals
A dyadic interval is a bounded real interval whose endpoints are and where and are integers. Depending on the context, either endpoint may or may not be included in the interval.
Dyadic intervals have the following properties:
- The length of a dyadic interval is always an integer power of two.
- Each dyadic interval is contained in exactly one dyadic interval of twice the length.
- Each dyadic interval is spanned by two dyadic intervals of half the length.
- If two open dyadic intervals overlap, then one of them is a subset of the other.
The dyadic intervals consequently have a structure that reflects that of an infinite binary tree.
Dyadic intervals are relevant to several areas of numerical analysis, including
Generalizations
Balls
An open finite interval is a 1-dimensional open
If a half-space is taken as a kind of degenerate ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.
Multi-dimensional intervals
A finite interval is (the interior of) a 1-dimensional hyperrectangle. Generalized to real coordinate space an axis-aligned hyperrectangle (or box) is the Cartesian product of finite intervals. For this is a rectangle; for this is a
Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any intervals, is sometimes called an -dimensional interval.[citation needed]
A facet of such an interval is the result of replacing any non-degenerate interval factor by a degenerate interval consisting of a finite endpoint of The faces of comprise itself and all faces of its facets. The corners of are the faces that consist of a single point of [citation needed]
Convex polytopes
Any finite interval can be constructed as the intersection of half-bounded intervals (with an empty intersection taken to mean the whole real line), and the intersection of any number of half-bounded intervals is a (possibly empty) interval. Generalized to -dimensional affine space, an intersection of half-spaces (of arbitrary orientation) is (the interior of) a convex polytope, or in the 2-dimensional case a convex polygon.
Domains
An open interval is a connected open set of real numbers. Generalized to topological spaces in general, a non-empty connected open set is called a domain.
Complex intervals
Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.[12]
Intervals in posets and preordered sets
Definitions
The concept of intervals can be defined in arbitrary
where means Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
defined by adding new smallest and greatest elements (even if there were ones), which are subsets of In the case of one may take to be the
Convex sets and convex components in order theory
A subset of the
is convex, but not an interval of since there is no square root of two in
Let be a
Properties
A generalization of the characterizations of the real intervals follows. For a non-empty subset of a linear continuum the following conditions are equivalent.[16]: 153, Theorem 24.1
- The set is an interval.
- The set is order-convex.
- The set is a connected subset when is endowed with the order topology.
For a subset of a
Applications
In general topology
Every
The concepts of convex sets and convex components are used in a proof that every
Topological algebra
This section needs additional citations for verification. (September 2023) |
Intervals can be associated with points of the plane, and hence regions of intervals can be associated with
The direct sum algebra has two
Every interval can be considered a symmetric interval around its midpoint. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" [x, −x] is used along with the axis of intervals [x, x] that reduce to a point. Instead of the direct sum the ring of intervals has been identified
where
This linear mapping of the plane, which amounts of a
See also
- Arc (geometry)
- Inequality
- Interval graph
- Interval finite element
- Interval (statistics)
- Line segment
- Partition of an interval
- Unit interval
References
- ISBN 1-886529-02-7.
- ^ ISBN 0-7637-1497-6.
- ^ Weisstein, Eric W. "Interval". mathworld.wolfram.com. Retrieved 2020-08-23.
- ^ "Interval and segment", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- LCCN 2016940817. See Definition 9.1.1.
- ISBN 0691005478.
- ^ "Interval and segment - Encyclopedia of Mathematics". encyclopediaofmath.org. Archived from the original on 2014-12-26. Retrieved 2016-11-12.
- ISBN 0-07-054235-X.
- ^ "Why is American and French notation different for open intervals (x, y) vs. ]x, y[?". hsm.stackexchange.com. Retrieved 28 April 2018.
- ^ Tao (2016), p. 214, See Lemma 9.1.12.
- S2CID 16796699. Retrieved 2012-04-05.
- ISBN 978-3-527-40134-5
- Zbl 1080.91001.
- ^ Zbl 0269.54009.
- ^ Zbl 0189.53103.
- Zbl 0951.54001.
- Zbl 0684.54001.
- ^ Kaj Madsen (1979) Review of "Interval analysis in the extended interval space" by Edgar Kaucher[permanent dead link] from Mathematical Reviews
- ^ D. H. Lehmer (1956) Review of "Calculus of Approximations"[permanent dead link] from Mathematical Reviews
Bibliography
- T. Sunaga, "Theory of interval algebra and its application to numerical analysis" Archived 2012-03-09 at the Wayback Machine, In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp. 126–143.
External links
- A Lucid Interval by Brian Hayes: An American Scientist article provides an introduction.
- Interval computations website Archived 2006-03-02 at the Wayback Machine
- Interval computations research centers Archived 2007-02-03 at the Wayback Machine
- Interval Notation by George Beck, Wolfram Demonstrations Project.
- Weisstein, Eric W. "Interval". MathWorld.