
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a real interval is the
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 all
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative
infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, indicating the interval extends without a
bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.
For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
; the set of all
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 fo ...
is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted .
Intervals are ubiquitous in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
. For example, they occur implicitly in the
epsilon-delta definition of continuity; the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two imp ...
asserts that the image of an interval by a
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
is an interval;
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s of
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
s are defined over an interval; etc.
Interval arithmetic
Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numeri ...
consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of
input data and
rounding error
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Roun ...
s.
Intervals are likewise defined on an arbitrary
totally ordered
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( r ...
set, such as
integers
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
or
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
. The notation of integer intervals is considered
in the special section below.
Definitions and terminology
An ''interval'' is a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s that contains all real numbers lying between any two numbers of the subset. In particular, the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
and the entire set of real numbers
are both intervals.
The ''endpoints'' of an interval are its
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
, and its
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
, if they exist as real numbers.
If the infimum does not exist, one says often that the corresponding endpoint is
Similarly, if the supremum does not exist, one says that the corresponding endpoint is
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
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if ever ...
of the real numbers. This characterization is used to specify intervals by means of ', which is described below.
An ' does not include any endpoint, and is indicated with parentheses.
For example,
is the interval of all real numbers greater than and less than . (This interval can also be denoted by , see below). The open interval consists of real numbers greater than , i.e., positive real numbers. The open intervals have thus one of the forms
:
where
and
are real numbers such that
In the last case, the resulting interval is the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
and does not depend on . The open intervals are those intervals that are
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s for the usual
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on the real numbers.
A ' is an interval that includes all its endpoints and is denoted with square brackets.
[ For example, means greater than or equal to and less than or equal to . Closed intervals have one of the following forms in which and are real numbers such that
:
The closed intervals are those intervals that are ]closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s for the usual topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on the real numbers.
A ' 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. For example, means greater than and less than or equal to , while means greater than or equal to and less than . The half-open intervals have the form
:
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. The only intervals that appear twice in the above classification are and that are both open and closed.[ See Definition 9.1.1.]
A ' is any singleton set, 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
In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in ...
s, in the sense that their diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
(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, and is a special case of the Lp distance fo ...
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''-dim ...
) 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
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
(an element that is smaller than all other elements); ''right-open'' if it contains no maximum
In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (ma ...
; 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, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s of the real line in its standard topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, 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 (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
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 a ''subinterval'' of interval if is a subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of . An interval is a ''proper subinterval'' of if is a proper subset
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
of .
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'' 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 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, such as ...
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
An international standard is a technical standard developed by one or more international standards organizations. International standards are available for consideration and use worldwide. The most prominent such organization is the International O ...
ISO 31-11
ISO 31-11:1992 was the part of international standard ISO 31 that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000 ...
. Thus, in set builder notation
In mathematics and more specifically in set theory, set-builder notation is a notation for specifying a set by a property that characterizes its members.
Specifying sets by member properties is allowed by the axiom schema of specification. This ...
,
:
Each interval , , and represents the empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
, 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 mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
in set theory, the coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
of a point or vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
in analytic geometry
In 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 engineering, and als ...
and linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
, or (sometimes) a complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
in algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. That is why Bourbaki Bourbaki(s) may refer to :
Persons and science
* Charles-Denis Bourbaki (1816–1897), French general, son of Constantin Denis Bourbaki
* Colonel Constantin Denis Bourbaki (1787–1827), officer in the Greek War of Independence and serving in the ...
introduced the notation to denote the open interval. The notation too is occasionally used for ordered pairs, especially in computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied 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
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
, 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 fo ...
, also written as The context affects some of the above definitions and terminology. For instance, the interval = is closed in the realm of ordinary reals, but not in the realm of the extended reals.
Integer intervals
When and are integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, 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 language
A programming language is a system of notation for writing computer programs.
Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
s; in Pascal, 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 ...
.
Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis
The ellipsis (, plural ellipses; from , , ), rendered , alternatively described as suspension points/dots, points/periods of ellipsis, or ellipsis points, or colloquially, dot-dot-dot,. According to Toner it is difficult to establish when t ...
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 , , or . Alternate-bracket notations like or are rarely used for integer intervals.
Properties
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 It follows that the image of an interval by any continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
from to is also an interval. This is one formulation of the intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two imp ...
.
The intervals are also the convex subset
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s of The interval enclosure of a subset is also the convex hull
In geometry, the convex hull, 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 space, ...
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
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is a connected subset.) In other words, we have
:
:
:
:
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 metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, its open 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 defin ...
s are the open bounded intervals , 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 defin ...
s are the closed bounded intervals . In particular, the metric
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
and order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
topologies in the real line coincide, which is the standard topology of the real line.
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 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 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
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...
.
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 analysis, multiresolution methods, v ...
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 n ...
. Another way to represent such a structure is p-adic analysis
In mathematics, ''p''-adic analysis is a branch of number theory that studies functions of ''p''-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex ...
(for ).
Generalizations
Balls
An open finite interval is a 1-dimensional open ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
with a center at and a radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
of The closed finite interval , b
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is the corresponding closed ball, and the interval's two endpoints form a 0-dimensional sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. Generalized to -dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called a disk.
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
In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient cond ...
. Generalized to real coordinate space
In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''.
...
an axis-aligned hyperrectangle (or box) is the Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of finite intervals. For this is a rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
; for this is a rectangular cuboid
A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped.
Many writers just call these ...
(also called a "box
A box (plural: boxes) is a container with rigid sides used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides (typically rectangular prisms). Boxes can be very small (like a matchbox) or v ...
").
Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any intervals, is sometimes called an -dimensional interval.
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
Convex polytopes
Any finite interval can be constructed as the intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
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
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
, an intersection of half-spaces (of arbitrary orientation) is (the interior of) a convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
, or in the 2-dimensional case a convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...
.
Domains
An open interval is a connected open set of real numbers. Generalized to topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s in general, a non-empty connected open set is called a domain.
Complex intervals
Intervals of complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s can be defined as regions of the complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, either rectangular
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or 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 circular.
Intervals in posets and preordered sets
Definitions
The concept of intervals can be defined in arbitrary partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s or more generally, in arbitrary preordered sets. For a preordered set and two elements one similarly defines the intervals
:
:
:
:
:(a,\infty) =\,
:[a,\infty) =\,
:(-\infty,b) =\,
:(-\infty,b =\,
:(-\infty,\infty) =X,
where x means x\lesssim y\not\lesssim x. Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
:\bar X=X\sqcup\
:-\infty
defined by adding new smallest and greatest elements (even if there were ones), which are subsets of X. In the case of X=\mathbb R one may take \bar\mathbb R to be the extended real line.
Convex sets and convex components in order theory
A subset A\subseteq X of the preordered set (X,\lesssim) is (order-)convex if for every x,y\in A and every x\lesssim z\lesssim y we have z\in A. Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in the totally ordered set
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( ref ...
(\mathbb Q,\le) of rational number
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 (for example,
The set of all ...
s, the set
:\mathbb Q=\
is convex, but not an interval of \mathbb Q, since there is no square root of two in \mathbb Q.
Let (X,\lesssim) be a preordered set and let Y\subseteq X. The convex sets of X contained in Y form a poset
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
under inclusion. A maximal element
In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an ...
of this poset is called a convex component of Y. By the Zorn lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (order theory), chain (that is, every total order, totally ordered subset) N ...
, any convex set of X contained in Y is contained in some convex component of Y, but such components need not be unique. In a totally ordered set
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( ref ...
, such a component is always unique. That is, the convex components of a subset of a totally ordered set form a partition.
Properties
A generalization of the characterizations of the real intervals follows. For a non-empty subset I of a linear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.
Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any t ...
(L,\le), the following conditions are equivalent.
* The set I is an interval.
* The set I is order-convex.
* The set I is a connected subset when L is endowed with the order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
.
For a subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
S of a lattice L, the following conditions are equivalent.
* The set S is a sublattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
and an (order-)convex set.
* There is an ideal I\subseteq L and a filter F\subseteq L such that S=I\cap F.
Applications
In general topology
Every Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any completely regular space that is also a ...
is embeddable into a product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
of the closed unit intervals ,1 Actually, every Tychonoff space that has a base of cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
\kappa is embeddable into the product ,1\kappa of \kappa copies of the intervals.
The concepts of convex sets and convex components are used in a proof that every totally ordered set
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( ref ...
endowed with the order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
is completely normal
Completely may refer to:
* ''Completely'' (Diamond Rio album)
* ''Completely'' (Christian Bautista album), 2005
* "Completely", a song by American singer and songwriter Michael Bolton
* "Completely", a song by Serial Joe from ''(Last Chance) A ...
[ or moreover, monotonically normal.][
]
Topological algebra
Intervals can be associated with points of the plane, and hence regions of intervals can be associated with region
In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...
s of the plane. Generally, an interval in mathematics corresponds to an ordered pair taken from the direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
\R \times \R of real numbers with itself, where it is often assumed that . For purposes of mathematical structure
In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...
, this restriction is discarded, and "reversed intervals" where are allowed. Then, the collection of all intervals can be identified with the topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
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. As an example, the direct sum of two abelian groups A and B is anothe ...
of \R with itself, where addition and multiplication are defined component-wise.
The direct sum algebra ( \R \oplus \R, +, \times) has two ideals, and . The identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of this algebra is the condensed interval . If interval is not in one of the ideals, then it has multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
. Endowed with the usual topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, the algebra of intervals forms a topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
. The group of units
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the ele ...
of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component
In mathematics, specifically group theory, the identity component of a group (mathematics) , group ''G'' (also known as its unity component) refers to several closely related notions of the largest connected space , connected subgroup of ''G'' co ...
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''-dim ...
. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" is used along with the axis of intervals 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 hyperbolic numbers by M. Warmus and D. H. Lehmer through the identification
:z = \tfrac12(x + y) + \tfrac12(x - y)j,
where j^2 = 1.
This linear mapping of the plane, which amounts of a ring isomorphism
In mathematics, 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 that preserves addition, multiplication and multiplicative identity ...
, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
.
See also
*Arc (geometry)
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
* Inequality
* Interval graph
*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)
In statistics, interval estimation is the use of Sample (statistics), sample data to estimation, estimate an ''interval (mathematics), interval'' of possible values of a Statistical parameter, parameter of interest. This is in contrast to point est ...
*Line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
*Partition of an interval
In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that
:.
In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itsel ...
*Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
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
The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
*
{{DEFAULTSORT:Interval (Mathematics)
Sets of real numbers
Order theory
Topology