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trichotomy-principle

Interval (mathematics)

In mathematics, an interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0le xle 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers R, the set of all positive real numbers, and the empty set.

Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then be extended to more complicated sets, leading to the Borel measure 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.

Notations for intervals

The interval of numbers between a and b, including a and b, is often denoted [a,b]. The two numbers are called the endpoints of the interval.

To indicate that one of the endpoints is to be excluded from the set, many writers substitute a parenthesis for the corresponding square bracket. Thus, in set builder notation,

(a,b)={xinR,|,a,
[a,b)={xinR,|,ale x,
(a,b]={xinR,|,a,
[a,b]={xinR,|,ale xle b}.
Note that (a,a), [a,a), and (a,a] denote the empty set, whereas [a,a] denotes the set {a}. When a > b, all four notations are usually assumed to represent the empty set.

International standard ISO 31-11 also defines another notation for intervals, which seems to be more commonly taught in European and South American secondary schools. It uses an inwards pointing bracket to indicate inclusion of the endpoint, and outwards-pointing bracket for exclusion:

left]a,bright[= {x,|, a< x < b}
left[a,bright[= {x,|, ale x < b}
left]a,bright] = {x,|, a< x le b}
[a,b] = { x ,| ,a le x le b }

Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a,b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. The notation [a,b] too is occasionally used for ordered pairs, especially in computer science.

Some authors use left]a,bright[ to denote the complement of the interval (a,b); namely, the set of all real numbers that are either {}leq a or {}geq b.

In countries where numbers are written with a decimal comma, a semicolon may be used as a separator, to avoid ambiguity.

Infinite endpoints

In both styles of notation, one may use an infinite endpoint to indicate that there is no bound in that direction. Specifically, one may use a=-infty or b=+infty (or both). For example, (0,+infty) is the set of all positive real numbers, and (-infty,+infty) is the set of real numbers.

The notations [-infty,b], [-infty,b), [a,+infty], and (a,+infty] are ambiguous. For authors who define intervals as subsets of the real numbers, those notations are either meaningless, or equivalent to the open variants. In the latter case, the interval comprising all real numbers is both open and closed, (-infty,+infty) = [-infty,+infty] = [-infty,+infty) = (-infty,+infty].

Terminology

A degenerate interval is any set consisting of a single real number. Some authors include the empty set in this definitions. An 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 bounded sets, in the sense that their diameter (the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, or size of the interval. The size of unboundded intervals is usually defined as +infty, and the size of the empty interval may be defined as 0 or left undefined.

The center of bounded interval with endpoints a and b is (a+b)/2, and its radius is the half-length left|a-bright|/2. These concepts are undefined for empty or unbounded intervals.

An interval is said to be left-open if and only if it has no minimum (an element that is smaller than all other elements); right-open if it has no maximum; and open if it has both properties. The interval [0,1) = {x,|,0leq x < 1}, for example, is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative reals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of the real line in its standard topology.

An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closed if it hs both. These definitions are usually extended to include the empty set and to the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed sets in that topology.

The interior of an interval I is the the largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints.

For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X and is not properly contained in any other interval.

Classification of intervals

The intervals of real rumbers can be classified into eleven different types, listed below; where a and b are real numbers, with a < b:

empty: [b,a] = (a,a) = [a,a) = (a,a] = { } = emptyset
degenerate: [a,a] = {a}
proper and bounded:
open: (a,b)={x,|,a
closed: [a,b]={x,|,aleq xleq b}
left-closed, right-open: [a,b)={x,|,a,leq x
left-open, right-closed: (a,b]={x,|,a
left-bounded and right-unbounded:
left-open: (a,infty)={x,|,x>a}
left-closed: [a,infty)={x,|,xgeq a}
left-unbounded and right-bounded:
right-open: (-infty,b)={x,|,x
right-closed: (-infty,b]={x,|,xleq b}
unbounded at both ends: (-infty,infty)=R

Intervals of the extended real line

In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with -infty and +infty.

In this interpretation, the notations [-infty,b], [-infty,b), [a,+infty], and (a,+infty] are all meaningful and distinct. In particular, (-infty,+infty) denotes the set of all ordinary real numbers, while [-infty,+infty] denotes the extended reals.

This choice affects some of the above defintions and terminology. For instance, the interval (-infty,+infty)=R is closed in the realm of ordinary reals, but not in the realm of the extended reals.

Properties of intervals

The intervals are precisly the connected 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 subsets of R. The interval enclosure of a subset Xsubseteq R is also the convex hull of X.

The intersection of any collection of intervals is always an interval. The union of two open intervals (or two closed intervals) is an interval if andonly only if they have a non-empty intersection.

If R is viewed as a metric space, its open balls are the open bounded sets (a,b), and its closed balls are the closed bounded sets [a,b].

Any element x of an interval I defines a partition of I into three disjoint intervals I_1,I_2,I_3: respectively, the elements of I that are less than x, the singleton [x,x] = {x}, and the elements that are greater than x. The parts I_1 and I_3 are both non-empty (and have non-empty interiors) if an only if x is in the interior of I. This is an interval version of the trichotomy principle.

Dyadic intervals

A dyadic interval is a bounded real interval whose endpoints are frac{j}{2^n} and frac{j+1}{2^n}, where j and n are integers. Depending on the context, either endpoint may or may not be included in the interval.

Dyadic intervals have some nice properties, such as the following:

  • The length of a dyadic interval is always an integer power of two.
  • Every dyadic interval is contained in exactly one "parent" dyadic interval of twice the length.
  • Every dyadic interval is spanned by two "child" dyadic intervals of half the length.
  • If two dyadic intervals overlap, then one of them must be a subset of the other.

The dyadic intervals thus have a structure very similar to an infinite binary tree.

Dyadic intervals are relevant to several areas of numerical analysis, includeing adaptive mesh refinement, multigrid methods, and wavelet analysis.

Intervals in order theory

In order theory, the concept of an interval can be extended to totally ordered sets. An interval in a totally ordered set (Omega,le) is a subset S of Omega such that whenever x and x,zin S, then also yin S. The notations (a,b), [a,b), etc. are also used in this context. Thus, [a,b)={zin Omega,|,ale z < b}, for example.

Multi-dimensional intervals

In many contexts, an n-dimensional interval is defined as a subset of R^n that is the Cartesian product of n intervals, I = I_1times I_2 times cdots times I_n, one on each coordinate axis.

For n=2, this generally defines a rectangle whose sides are parallel to the coordinate axes; for n=3, it defines an axis-aligned rectangular box.

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.

See also

References

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