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In mathematics, a (real) interval is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, and any
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
(set of one element). Real intervals play an important role in the theory of
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff. Intervals are likewise defined on an arbitrary totally ordered set, such as integers or
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
. The notation of integer intervals is considered in the special section below.


Terminology

An does not include its endpoints, and is indicated with parentheses. For example, means greater than and less than . This means . This interval can also be denoted by , see below. A is an interval which includes all its limit points, and is denoted with square brackets. For example, means greater than or equal to and less than or equal to . A includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. For example, means greater than and less than or equal to , while means greater than or equal to and less than . A is any set consisting of a single real number (i.e., an interval of the form ). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements. 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 set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of m ...
s, in the sense that their diameter (which is equal to the
absolute difference The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...
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 (or left undefined). The centre (
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...
) of a bounded interval with endpoints and is , and its radius is the half-length . These concepts are undefined for empty or unbounded intervals. An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it contains neither. The interval , for example, is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s of the real line in its standard topology, and form a base of the open sets. 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 set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s in that topology. The interior of an interval is the largest open interval that is contained in ; it is also the set of points in which are not endpoints of . The closure of is the smallest closed interval that contains ; which is also the set augmented with its finite endpoints. For any set of real numbers, the interval enclosure or interval span of is the unique interval that contains , and does not properly contain any other interval that also contains . An interval is subinterval of interval if is a subset of . An interval is a proper subinterval of if is a proper subset of .


Note on conflicting terminology

The terms segment and interval have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The ''Encyclopedia of Mathematics'' 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'' 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 and , including and , is often denoted . The two numbers are called the ''endpoints'' of the interval. In countries where numbers are written with a
decimal comma A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The cho ...
, a
semicolon The semicolon or semi-colon is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought. When a ...
may be used as a separator to avoid ambiguity.


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 International standard ISO 31-11. Thus, in
set builder notation In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Def ...
, : \begin a,b = \mathopena,b\mathclose &= \, \\ a,b = \mathopen a,b\mathclose &= \, \\ a,b = \mathopena,b\mathclose &= \, \\ a,b = \mathopen a,b\mathclose &= \. \end Each interval , , and represents the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, whereas denotes the singleton set . When , 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 is often used to denote an ordered pair in set theory, the
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
of a point or vector in
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineer ...
and linear algebra, or (sometimes) a complex number in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
. That is why Bourbaki introduced the notation to denote the open interval. The notation too is occasionally used for ordered pairs, especially in computer science. Some authors such as Yves Tillé use to denote the complement of the interval ; namely, the set of all real numbers that are either less than or equal to , or greater than or equal to .


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  ,  ,  , and 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 infinite endpoint to indicate that there is no bound in that direction. For example, is the set of
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
, also written as \mathbb_+. The context affects some of the above definitions and terminology. For instance, the interval  = \R is closed in the realm of ordinary reals, but not in the realm of the extended reals.


Integer intervals

When and are integers, the notation ⟦''a, b''⟧, or or or just , is sometimes used to indicate the interval of all ''integers'' between and included. The notation is used in some programming languages; in
Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
. 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  ,  , or . Alternate-bracket notations like or are rarely used for integer intervals.


Classification of intervals

The intervals of real numbers can be classified into the eleven different types listed below, where and are real numbers, and a < b: * Empty: ,a= (b,a) = ,a) = (b,a= (a,a) = ,a) = (a,a= \ = \varnothing * Degenerate: ,a= \ * Proper and bounded: ** Open: (a,b) = \ ** Closed: ,b= \ ** Left-closed, right-open: ,b) = \ ** Left-open, right-closed: (a,b= \ * Left-bounded and right-unbounded: ** Left-open: (a,+\infty) = \ ** Left-closed: ,+\infty) = \ * Left-unbounded and right-bounded: ** Right-open: (-\infty,b) = \ ** Right-closed: (-\infty,b= \ * Unbounded at both ends (simultaneously open and closed): (-\infty,+\infty) = \R:


Properties of intervals

The intervals are precisely the
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
subsets of \R. It follows that the image of an interval by any continuous function is also an interval. This is one formulation of the intermediate value theorem. The intervals are also the
convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
s of \R. The interval enclosure of a subset X\subseteq \R is also the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...
of X. 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 – e.g., (a,b) \cup ,c= (a,c]. If \R is viewed as a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, its open balls are the open bounded sets , and its
closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defi ...
s are the closed bounded sets . Any element  of an interval  defines a partition of  into three disjoint intervals 1, 2, 3: respectively, the elements of  that are less than , the singleton  ,x= \, and the elements that are greater than . The parts 1 and 3 are both non-empty (and have non-empty interiors), if and only if is in the interior of . This is an interval version of the trichotomy principle.


Dyadic intervals

A ''dyadic interval'' is a bounded real interval whose endpoints are \frac and \frac, where j and n 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 In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tr ...
. Dyadic intervals are relevant to several areas of numerical analysis, including
adaptive mesh refinement In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions ...
,
multigrid methods In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhi ...
and
wavelet analysis A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the nu ...
. Another way to represent such a structure is p-adic analysis (for ).


Generalizations


Multi-dimensional intervals

In many contexts, an n-dimensional interval is defined as a subset of \R^n that is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of n intervals, I = I_1\times I_2 \times \cdots \times I_n, one on each
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
axis. For n=2, this can be thought of as region bounded by a square or rectangle, whose sides are parallel to the coordinate axes, depending on whether the width of the intervals are the same or not; likewise, for n=3, this can be thought of as a region bounded by an axis-aligned cube or a
rectangular cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
. In higher dimensions, the Cartesian product of n intervals is bounded by an
n-dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
hypercube or hyperrectangle. A facet of such an interval I is the result of replacing any non-degenerate interval factor I_k by a degenerate interval consisting of a finite endpoint of I_k. The faces of I comprise I itself and all faces of its facets. The corners of I are the faces that consist of a single point of \R^n.


Complex intervals

Intervals of complex numbers can be defined as regions of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
, either
rectangular In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
or
circular Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
.


Topological algebra

Intervals can be associated with points of the plane, and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair (''x,y'') taken from the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
R × R of real numbers with itself, where it is often assumed that ''y'' > ''x''. For purposes of
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
, this restriction is discarded, and "reversed intervals" where ''y'' − ''x'' < 0 are allowed. Then, the collection of all intervals 'x,y''can be identified with the topological ring formed by the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of R with itself, where addition and multiplication are defined component-wise. The direct sum algebra ( R \oplus R, +, \times) has two
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
s, and . The
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
of this algebra is the condensed interval ,1 If interval 'x,y''is not in one of the ideals, then it has multiplicative inverse /''x'', 1/''y'' Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of this group is quadrant I. Every interval can be considered a symmetric interval around its
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...
. 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 R \oplus R, the ring of intervals has been identified D. H. Lehmer (1956
Review of "Calculus of Approximations"
from Mathematical Reviews
with the split-complex number plane by M. Warmus and D. H. Lehmer through the identification : ''z'' = (''x'' + ''y'')/2 + j (''x'' − ''y'')/2. This linear mapping of the plane, which amounts of a
ring isomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.


See also

*
Arc (geometry) In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
*
Inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
*
Interval graph In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. In ...
*
Interval finite element In numerical analysis, the interval finite element method (interval FEM) is a finite element method that uses interval parameters. Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of ...
* Interval (statistics) * Line segment * Partition of an interval * Unit interval


References


Bibliography

* T. Sunaga
"Theory of interval algebra and its application to numerical analysis"
, 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: A
American Scientist article
provides an introduction.




Interval Notation
by George Beck, Wolfram Demonstrations Project. * {{DEFAULTSORT:Interval (Mathematics) Sets of real numbers Order theory Topology