The additionx +a on the number line. All numbers greater thanx and less thanx +a fall within that open interval.Numeric intervals on the positive and negative sides of thenumber line.
Inmathematics, areal interval is theset of allreal numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negativeinfinity, indicating the interval extends without abound. 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 of0,1, and all numbers in between is an interval, denoted[0, 1] and called theunit interval; the set of allpositive real numbers is an interval, denoted(0, ∞); the set of all real numbers is an interval, denoted(−∞, ∞); and any single real numbera is an interval, denoted[a,a].
Interval arithmetic 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 ofinput data androunding errors.
Unless explicitly otherwise specified, all intervals considered in this article are real intervals, that is, intervals of real numbers. Notable generalizations are summarized in a section below possibly with links to separate articles.
Aninterval is asubset of thereal numbers that contains all real numbers lying between any two numbers of the subset. Examples are the numbers from one to two,, and the numbers greater than 10, i.e.. In particular, theempty set and the entire set of real numbers are both intervals.[1]
Theendpoints of an interval are itssupremum (least upper bound), and itsinfimum (greatest lower bound), if they exist as real numbers.[1] If the infimum does not exist and the interval is not empty, one says often that the corresponding endpoint is negative infinity, written Similarly, if the supremum of a non-empty interval does not exist, one says that the corresponding endpoint is positive infinity, written
Non-empty intervals are completely determined by their endpoints and whether each endpoint belongs to the interval. This is a consequence of theleast-upper-bound property of the real numbers, which implies that if the elements of a non-empty interval are all less than some finite value, then the interval has a supremum. This characterization is used to specify intervals by means ofinterval notation, where a square or rounded bracket (parenthesis) indicates whether or not an endpoint belongs to the inteval.
Anopen interval does not include any endpoint and can be succinctly indicated with parentheses.[2] 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 theempty set and does not depend on. The open intervals are those intervals that areopen sets for the usualtopology on the real numbers, and they form abase of the open sets.
Aclosed interval is an interval that includes all its finite endpoints. When both endpoints are finite, they are enclosed in square brackets.[2] For example,[0, 1] is the closed interval with contents greater than or equal to0 and less than or equal to1. Closed intervals, other than the empty interval, have one of the following forms in whicha andb are real numbers such that
The closed intervals are those intervals that areclosed sets for the usualtopology on the real numbers.
Ahalf-open interval has two distinct finite endpoints, and includes one but not the other. It is said to beleft-open orright-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 than0 and less than or equal to1, while[0, 1) means greater than or equal to0 and less than1. 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.[4][5]
Adegenerate interval is anyset consisting of a single real number (i.e., an interval of the form[a,a]).[6] Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to beproper, and has infinitely many elements.
An interval is said to beleft-bounded orright-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to bebounded, if it is both left- and right-bounded; and is said to beunbounded otherwise. Intervals that are bounded at only one end are said to behalf-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 asfinite intervals.
Bounded intervals arebounded sets, in the sense that theirdiameter (which is equal to theabsolute difference between the endpoints) is finite. The diameter may be called thelength,width,measure,range, orsize of the interval. The size of unbounded intervals is usually defined as+∞, and the size of the empty interval may be defined as0 (or left undefined).
Thecentre (midpoint) of a bounded interval with endpointsa andb is(a + b)/2, and itsradius is the half-length|a − b|/2. These concepts are undefined for empty or unbounded intervals.
An interval is said to beleft-open if and only if it contains nominimum (an element that is smaller than all other elements);right-open if it contains nomaximum; andopen if it contains neither. The interval[0, 1) = {x | 0 ≤x < 1}, for example, is left-closed and right-open. The set of non-negative reals is a closed interval that is right-open but not left-open.
An interval is said to beleft-closed if it has a minimum element or is left-unbounded,right-closed if it has a maximum or is right unbounded; it is simplyclosed if it is both left-closed and right closed.
An intervalI is asubinterval of intervalJ ifI is asubset ofJ. An intervalI is aproper subinterval ofJ ifI is aproper subset ofJ.
Theinterior of an intervalI is the largest open interval that is contained inI; it is also the set of points inI which are not endpoints ofI. Theclosure ofI is the smallest closed interval that containsI; which is also the setI augmented with its finite endpoints.
For any setX of real numbers, theinterval enclosure orinterval span ofX is the unique interval that containsX, and does not properly contain any other interval that also containsX.
There is conflicting terminology for the termssegment andinterval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. TheEncyclopedia of Mathematics[7] definesinterval (without a qualifier) to exclude both endpoints (i.e., open interval) andsegment to include both endpoints (i.e., closed interval), while Rudin'sPrinciples 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 terminterval (qualified byopen,closed, orhalf-open), regardless of whether endpoints are included.
The interval of numbers betweena andb, includinga andb, is often denoted[a, b]. The two numbers are called theendpoints of the interval. In countries where numbers are written with adecimal comma, asemicolon may be used as a separator to avoid ambiguity.
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 inInternational standardISO 31-11. Thus, inset builder notation,
Each interval(a, a),[a, a), and(a, a] represents theempty set, whereas[a, a] denotes the singleton set {a}. Whena >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 anordered pair in set theory, thecoordinates of apoint orvector inanalytic geometry andlinear algebra, or (sometimes) acomplex number inalgebra. That is whyBourbaki introduced the notation]a,b[ to denote the open interval.[9] The notation[a,b] too is occasionally used for ordered pairs, especially incomputer science.
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 toa, or greater than or equal tob.
In some contexts, an interval may be defined as a subset of theextended 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 aninfinite endpoint to indicate that there is no bound in that direction. For example,(0, +∞) is the set ofpositive real numbers, 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.
Whena andb areintegers, the notations ⟦a, b⟧,[a ..b],{a ..b}, or justa ..b, are sometimes used to indicate the interval of allintegers betweena andb included. The notation[a ..b] is used in someprogramming languages; inPascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of validindices of anarray.
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 writinga ..b − 1 ,a + 1 ..b , ora + 1 ..b − 1. Alternate-bracket notations like[a ..b) or[a ..b[ are rarely used for integer intervals.[citation needed]
The intervals are also theconvex subsets of The interval enclosure of a subset is also theconvex hull of
Theclosure 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 everyconnected subset of atopological 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 ametric space, itsopen balls are the open bounded intervals (c + r, c − r), and itsclosed balls are the closed bounded intervals [c + r, c − r]. In particular, themetric andorder topologies in the real line coincide, which is the standard topology of the real line.
Any element x of an interval I defines a partition of I into three disjoint intervalsI1, I2, I3: respectively, the elements of I that are less than x, the singleton and the elements that are greater than x. The partsI1 andI3 are both non-empty (and have non-empty interiors), if and only ifx is in the interior of I. This is an interval version of thetrichotomy principle.
Adyadic 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 integerpower 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 infinitebinary tree.
Intervals are ubiquitous inmathematical analysis, where they are used to express ideas and often occur in key results.
Theintegral of areal function is defined over an interval. The endpoints of the interval involved usually occur as a subscript and superscript, so the integral applies to all belonging to the interval.
Intervals occur implicitly in theepsilon-delta definition of continuity of a function: the following account makes them explicit. The function is said to be continuous at a point if for any given value (epsilon greater than zero) there is a value (delta greater than zero) for which lies in the open interval whenever is chosen from the interval. The possible values of and themselves belong to the unbounded interval, but are usually considered to describe small positive increments. The idea is that a small symmetric interval around point exists where the value of stays within an open interval of radius centred around.
Theintermediate value theorem captures the intuition that if is a real valuedcontinuous function on an interval and is any value between and, then we expect to find a value between and where. For example, if is defined on the interval, then given between and, there is a value between and where.An equivalent formulation of the theorem asserts that theimage of an interval by a continuous function is an interval.
When sampling is repeated there is a pre-set probability, known as theconfidence level, that a corresponding interval contains the true value of the unknown parameter. For example, if the chosen confidence level were 0.95 and the same sampling procedure were repeated many times, in the long run approximately 95% of the resulting intervals would be expected to contain the true value.
Thenormal distribution provides a simplified illustration. It has aprobability density function whose graph is the familiar bell curve. The peak occurs at its mean (the Greek letter mu) and its width can be described by its standard deviation (sigma). These two parameters distinguish one bell curve from another, but in all cases the region within two standard deviations either side of the mean represents a probability of approximately 0.95. This can be written
for a normally distributed random variable.
Take to be thesample mean for a fixed sample size, which is anestimator for. It also has a normal distribution with its own standard deviation. The previous inequalities can then be written in terms of to give
If the value of the standard deviation is known, then the interval will be a confidence interval for with a confidence level of approximately 0.95. Its endpoints are the random variables and, whose actual values will depend on the sample taken.
EveryTychonoff space is embeddable into aproduct space of the closed unit intervals Actually, every Tychonoff space that has abase ofcardinality is embeddable into the product of copies of the intervals.[12]: p. 83, Theorem 2.3.23
An open finite interval is a 1-dimensional openball with acenter at and aradius of The closed finite interval is the corresponding closed ball, and the interval's two endpoints form a 0-dimensionalsphere. Generalized to-dimensionalEuclidean space, 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 adisk.
If ahalf-space is taken as a kind ofdegenerate 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.
Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any intervals, is sometimes called an-dimensional interval.[citation needed]
Afacet of such an interval is the result of replacing any non-degenerate interval factor by a degenerate interval consisting of a finite endpoint of Thefaces of comprise itself and all faces of its facets. Thecorners of are the faces that consist of a single point of[citation needed]
Any finite interval can be constructed as theintersection 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-dimensionalaffine space, an intersection of half-spaces (of arbitrary orientation) is (the interior of) aconvex polytope, or in the 2-dimensional case aconvex polygon.
An open interval is a connected open set of real numbers. Generalized totopological spaces in general, a non-empty connected open set is called adomain.
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 theextended real line.
A subset of thepreordered set is(order-)convex if for every and every we have Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in thetotally ordered set ofrational numbers, the set
is convex, but not an interval of since there is no square root of two in
Let be apreordered set and let The convex sets of contained in form aposet under inclusion. Amaximal element of this poset is called aconvex component of[14]: Definition 5.1 [13]: 727 By theZorn lemma, any convex set of contained in is contained in some convex component of but such components need not be unique. In atotally ordered set, such a component is always unique. That is, the convex components of a subset of a totally ordered set form apartition.
A generalization of the characterizations of the real intervals follows. For a non-empty subset of alinear continuum the following conditions are equivalent.[17]: 153, Theorem 24.1
The set is an interval.
The set is order-convex.
The set is a connected subset when is endowed with theorder topology.
For asubset of alattice the following conditions are equivalent.
Intervals can be associated with points of the plane, and hence regions of intervals can be associated withregions of the plane. Generally, an interval in mathematics corresponds to an ordered pair(x,y) taken from thedirect product of real numbers with itself, where it is often assumed thaty >x. For purposes ofmathematical structure, this restriction is discarded,[18] and "reversed intervals" wherey −x < 0 are allowed. Then, the collection of all intervals[x,y] can be identified with thetopological ring formed by thedirect sum of with itself, where addition and multiplication are defined component-wise.
The direct sum algebra has twoideals, { [x,0] :x ∈ R } and { [0,y] :y ∈ R }. Theidentity element of this algebra is the condensed interval[1, 1]. If interval[x,y] is not in one of the ideals, then it hasmultiplicative inverse[1/x, 1/y]. Endowed with the usualtopology, the algebra of intervals forms atopological ring. Thegroup of units of this ring consists of fourquadrants determined by the axes, or ideals in this case. Theidentity component of this group is quadrant I.
Every interval can be considered a symmetric interval around itsmidpoint. 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[19] with thehyperbolic numbers by M. Warmus andD. H. Lehmer through the identification
where
This linear mapping of the plane, which amounts of aring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such aspolar decomposition.
^Engelking, Ryszard (1989).General topology. Sigma Series in Pure Mathematics. Vol. 6 (Revised and completed ed.). Berlin: Heldermann Verlag.ISBN3-88538-006-4.MR1039321.Zbl0684.54001.
