Interval (mathematics)

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The addition x + a on the number line. All numbers greater than x and less than x + a fall within that open interval. Interval0.png
The addition x + a on the number line. All numbers greater than x and less than x + a fall within that open interval.

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).

Contents

Real 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 of real numbers, 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.

Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.

Terminology

An open interval does not include its endpoints, and is indicated with parentheses. [1] [2] For example, (0,1) means greater than 0 and less than 1. This means (0,1) = {x | 0 <x< 1}.

A closed interval is an interval which includes all its limit points, and is denoted with square brackets. [1] [2] For example, [0,1] means greater than or equal to 0 and less than or equal to 1.

A half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. [3] For example, (0,1] means greater than 0 and less than or equal to 1, while [0,1) means greater than or equal to 0 and less than 1.

A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a,a]). [3] 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 sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined).

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

An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it has both properties. The interval [0,1) = {x | 0 ≤ 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, is a right-open but not left-open interval. The open intervals are open sets 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, right-closed if it has a maximum, and simply closed if it has both. These definitions are usually extended to include the empty set and 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 largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints.

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

An interval I is subinterval of interval J if I is a subset of J. An interval I is a proper subinterval of J if I is a proper subset of J.

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 [4] 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 [5] calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half-open), regardless of whether endpoints are included.

Notations for intervals

The interval of numbers between a and b, including a and b, is often denoted [a, b]. [1] The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon 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,

Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the singleton set {a}. When a > b, all four notations are usually taken to represent the empty set.

Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a, b) is often used to denote an 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. That is why Bourbaki introduced the notation ]a, b[ to denote the open interval. [6] The notation [a, b] too is occasionally used for ordered pairs, especially in computer science.

Some authors use ]a, b[ to denote the complement of the interval (a, b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.

Infinite endpoints

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

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

Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, (0, +∞) is the set of positive real numbers, also written as . [7] 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 a and b are integers, the notation ⟦a, b⟧, or [a .. b] or {a .. b} or just a .. b, is sometimes used to indicate the interval of all integers between a and b included. The notation [a .. b] is used in some programming languages; 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 integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations like [a .. b) or [a .. b[ are rarely used for integer intervals.[ citation needed ]

Classification of intervals

The intervals of real numbers can be classified into the eleven different types listed below[ citation needed ], where a and b are real numbers, and :

Empty:
Degenerate:
Proper and bounded:
Open:
Closed:
Left-closed, right-open:
Left-open, right-closed:
Left-bounded and right-unbounded:
Left-open:
Left-closed:
Left-unbounded and right-bounded:
Right-open:
Right-closed:
Unbounded at both ends (simultaneously open and closed): :

Properties of intervals

The intervals are precisely the connected subsets of . 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 . The interval enclosure of a subset is also the convex hull of .

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., ).

If is viewed as a metric space, its open balls are the open bounded sets (c + r, c − r), and its closed balls are the closed bounded sets [c + r, c − r].

Any element x of an interval I defines a partition of I into three disjoint intervals I1, I2, I3: respectively, the elements of I that are less than x, the singleton , and the elements that are greater than x. The parts I1 and I3 are both non-empty (and have non-empty interiors), if and only if x is in the interior of I. This is an interval version of the 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 dyadic intervals consequently have a structure that reflects that of an infinite binary tree.

Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for p = 2). [8]

Generalizations

Multi-dimensional intervals

In many contexts, an -dimensional interval is defined as a subset of that is the Cartesian product of intervals, , one on each coordinate axis.

For , 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 , this can be thought of as a region bounded by an axis-aligned cube or a rectangular cuboid. In higher dimensions, the Cartesian product of intervals is bounded by an n-dimensional hypercube or hyperrectangle.

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 .

Complex intervals

Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular. [9]

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 R × R of real numbers with itself, where it is often assumed that y > x. For purposes of mathematical structure, this restriction is discarded, [10] and "reversed intervals" where yx < 0 are allowed. Then, the collection of all intervals [x,y] can be identified with the topological ring formed by the direct sum of R with itself, where addition and multiplication are defined component-wise.

The direct sum algebra has two ideals, { [x,0] : x ∈ R } and { [0,y] : y ∈ R }. The identity element of this algebra is the condensed interval [1,1]. If interval [x,y] is not in one of the ideals, then it has multiplicative inverse [1/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 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 [11] with the split-complex number plane by M. Warmus and D. H. Lehmer through the identification

z = (x + y)/2 + j (xy)/2.

This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.

See also

Related Research Articles

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Real analysis Mathematics of real numbers and real functions

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

Sequence Finite or infinite ordered list of elements

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

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<i>p</i>-adic number Number system for a prime p which extends the rationals, defining closeness differently

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

Dyadic rational Fraction with a power of two as its denominator

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A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

Function (mathematics) Mapping that associates a single output value to each input

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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.

In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.

In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Additionally, there are several uses and meanings for the various brackets.

Real number Number representing a continuous quantity

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".

In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.

In mathematics, the set of positive real numbers, is the subset of those real numbers that are greater than zero. The non-negative real numbers, also include zero. Although the symbols and are ambiguously used for either of these, the notation or for and or for has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.

References

  1. 1 2 3 "List of Arithmetic and Common Math Symbols". Math Vault. 2020-03-17. Retrieved 2020-08-23.
  2. 1 2 "Intervals". www.mathsisfun.com. Retrieved 2020-08-23.
  3. 1 2 Weisstein, Eric W. "Interval". mathworld.wolfram.com. Retrieved 2020-08-23.
  4. "Interval and segment - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Archived from the original on 2014-12-26. Retrieved 2016-11-12.
  5. Rudin, Walter (1976). Principles of Mathematical Analysis . New York: McGraw-Hill. pp.  31. ISBN   0-07-054235-X.
  6. "Why is American and French notation different for open intervals (x, y) vs. ]x, y[?". hsm.stackexchange.com. Retrieved 28 April 2018.
  7. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-23.
  8. Kozyrev, Sergey (2002). "Wavelet theory as p-adic spectral analysis". Izvestiya RAN. Ser. Mat. 66 (2): 149–158. arXiv: math-ph/0012019 . Bibcode:2002IzMat..66..367K. doi:10.1070/IM2002v066n02ABEH000381. S2CID   16796699 . Retrieved 2012-04-05.
  9. Complex interval arithmetic and its applications, Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, ISBN   978-3-527-40134-5
  10. Kaj Madsen (1979) Review of "Interval analysis in the extended interval space" by Edgar Kaucher [ permanent dead link ] from Mathematical Reviews
  11. D. H. Lehmer (1956) Review of "Calculus of Approximations" [ permanent dead link ] from Mathematical Reviews

Bibliography