Lihat sumber Selang (matematika)

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'''Selang''' ([[bilangan real]]) dalam [[matematika]] adalah suatu [[himpunan]] [[bilangan real]] dengan sifat bahwa setiap bilangan yang terletak di antara dua bilangan dalam himpunan itu juga termasuk ke dalam himpunan. Misalnya, himpunan semua bilangan {{mvar|x}} memenuhi {{math|0 ≤ ''x'' ≤ 1}} adalah suatu selang yang memuat {{math|0}} dan {{math|1}}, maupun semua bilangan di antara keduanya. Contoh lain selang adalah suatu himpunan dari semua bilangan real <math>\R</math>, himpunan semua bilangan real negatif, dan [[himpunan kosong]].

Selang real berperang penting dalam teori [[Integral|integrasi]], karena merupakan himpunan-himpunan paling sederhana yang "ukuran" atau "pengukuran" atau "panjang"-nya mudah didefinisikan. Konsep pengukuran dapat diperluas untuk himpunan-himpunan bilangan real yang lebih rumit, mengarah kepada [[ukuran Borel]] dan akhirnya kepada [[ukuran Lebesgue]].

Selang adalah pusat bagi [[aritmetika selang]], suatu teknik [[Metode numerik|komputasi numerik]] umum yang secara otomatis menyediakan penutupan pasti bagi rumus-rumus sembarang, bahkan dengan adanya ketidakpastian, perkiraan matematika, dan [[Galat pembulatan|pembulatan aritmetika]].
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Intervals are likewise defined on an arbitrary [[total order|totally ordered]] set, such as [[integers]] or [[rational numbers]]. The notation of integer intervals is considered [[#Integer intervals|in the special section below]].
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== Notasi untuk selang ==
Selang angka-angka antara {{mvar|a}} dan {{mvar|b}}, termasuk {{mvar|a}} dan {{mvar|b}}, sering dilambangkan dengan {{tutup-tutup|''a'', ''b''}}. Dua bilangan itu disebut "titik-titik ujung" (''endpoints'') suatu selang. Di negara-negara di mana [[bilangan desimal]] ditulis menggunakan [[tanda koma]], [[tanda titik koma]] dapat digunakan sebagai pemisah, untuk menghindari kerancuan.

=== Termasuk atau tidak termasuk titik ujung ===
Untuk mengindikasikan bahwa satu dari titik-titik ujung tidak disertakan dalam himpunan, tanda kurung siku dapat diganti dengan tanda kurung, atau sebaliknya. Kedua notasi ini dijelaskan dalam [[International standard]] [[ISO 31-11]]. Jadi, dalam [[notasi ungkapan himpunan]],
: <math> \begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R\,|\,a<x<b\}, \\{}
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R\,|\,a\le x<b\}, \\{}
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R\,|\,a<x\le b\}, \\{}
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R\,|\,a\le x\le b\}.
\end{align} </math>
Perhatikan bahwa {{buka-buka|''a'', ''a''}}, {{tutup-buka|''a'', ''a''}}, dan {{buka-tutup|''a'', ''a''}} melambangkan [[himpunan kosong]], sedangkan {{tutup-tutup|''a'', ''a''}} melanmbangkan himpunan&nbsp;{{math|{''a''} }}. Ketika {{math|''a'' > ''b''}}, maka keempat notasi ini biasanya diasumsikan melambangkan himpunan kosong.
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Both notations may overlap with other uses of parentheses and brackets in mathematics.  For instance, the notation <math>(a,b)</math> is often used to denote  an [[tuple|ordered pair]] in set theory, the [[coordinates]] of a [[point (geometry)|point]] or [[vector (mathematics)|vector]] in [[analytic geometry]] and [[linear algebra]], or (sometimes) a [[complex number]] in [[algebra]].  The notation <math>[a,b]</math> too is occasionally used for ordered pairs, especially in [[computer science]].

Some authors use <math>]a,b[</math> to denote the complement of the interval&nbsp;{{open-open|''a'', ''b''}}; namely, the set of all real numbers that are either less than or equal to {{mvar|a}}, or greater than or equal to {{mvar|b}}.

===Infinite endpoints===
In both styles of notation, one may use an [[infinity (mathematics)|infinite]] endpoint to indicate that there is no bound in that direction.  Specifically, one may use <math>a=-\infty</math> or <math>b=+\infty</math> (or both). For example, {{open-open|0, +∞}} is the set of all positive real numbers, and {{open-open|−∞, +∞}} is the set of real numbers.

The [[extended real number line]] includes {{math|−∞}} and {{math|+∞}} as elements. The notations {{closed-closed|−∞, ''b''}} , {{closed-open|−∞, ''b''}} , {{closed-closed|''a'', +∞}} , and {{open-closed|''a'', +∞}} may be used in this context. For example {{open-closed|−∞, +∞}} means the extended real numbers excluding only {{math|−∞}}.

===Integer intervals===
The notation {{closed-closed|''a'' .. ''b''}} when {{mvar|a}} and {{mvar|b}} are [[integer]]s, or {{math|{''a'' .. ''b''} }}, or just {{math|''a'' .. ''b''}} is sometimes used to indicate the interval of all ''integers'' between {{mvar|a}} and {{mvar|b}}, including both.  This notation is used in some [[programming language]]s; in [[Pascal programming language|Pascal]], for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid [[Indexed family|indices]] of an [[Array data type|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 {{math|''a'' .. ''b'' − 1}} ,  {{math|''a'' + 1 .. ''b''}} , or  {{math|''a'' + 1 .. ''b'' − 1}}.  Alternate-bracket notations like {{closed-open|''a'' .. ''b''}} or {{math|[''a'' .. ''b''[}} are rarely used for integer intervals{{citation needed|date=February 2014}}.

==Terminologi==
'''Interval terbuka''' (''open interval'') tidak menyertakan titik-titik ujung, dan diindikasikan dengan tanda kurung. Misalnya {{buka-buka|0,1}} berarti lebih besar dari {{math|0}} dan lebih kecil dari {{math|1}}.

'''Interval tertutup''' (''closed interval'') menyertakan titik-titik ujung, dan dilambangkan dengan tanda kurung siku. Misalnya {{tutup-tutup|0,1}} berarti lebih besar dari atau sama dengan {{math|0}} dan lebih kecil dari atau sama dengan {{math|1}}.

'''Interval degenerasi''' (''degenerate interval'') adalah [[:en:singleton set|himpunan yang terdiri dari satu bilangan real]]. Beberapa penulis menyertakan [[himpunan kosong]] dalam definisi ini. Suatu interval real yang tidak "kosong" maupun "degenerasi" dikatakan sebagai '''proper''', dan memiliki banyak elemen yang tak terhingga..

Suatu interval dikatakan '''berbatas kiri''' (''left-bounded'') atau '''berbatas kanan''' (''right-bounded'') jika ada sejumlah bilangan real yang masing-masing lebih kecil dari atau lebih besar dari semua elemen-elemennya. Suatu interval dikatakan '''berbatas''' (''bounded'') jika sekaligus berbatas kiri dan berbatas kanan; dan dikatakan '''tak berbatas''' (''unbounded'') jika sebaliknya.  Interval yang berbatas hanya pada satu sisi dikatakan sebagai '''berbatas setengah''' (''half-bounded''). [[Himpunan kosong]] adalah berbatas, dan himpunan semua bilangan real adalah satu-satunya interval yang tak berbatas pada kedua ujungnya. Interval berbatas umumnya disebut juga '''interval terhingga''' (''finite interval'').
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Bounded intervals are  [[bounded set]]s, 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''', or '''size''' of the interval.  The size of unbounded intervals is usually defined as {{math|+∞}}, and the size of the empty interval may be defined as {{math|0}} or left undefined.

The '''centre''' ([[midpoint]]) of bounded interval with endpoints {{mvar|a}} and {{mvar|b}} is {{math|(''a'' + ''b'')/2}}, and its '''radius''' is the half-length {{math|{{mabs|''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 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 {{closed-open|0,1}}&nbsp;= {{mset|''x'' | 0 ≤ ''x'' &lt; 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 set]]s of the real line in its standard [[point-set topology|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 has 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 set]]s in that topology.

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

For any set {{mvar|X}} of real numbers, the '''interval enclosure''' or '''interval span''' of {{mvar|X}} is the unique interval that contains {{mvar|X}} and does not properly contain any other interval that also contains {{mvar|X}}.
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== Penggolongan selang ==
Selang bilangan real dapat digolongkan ke dalam 11 jenis yang berbeda, di mana {{mvar|a}} dan {{mvar|b}} adalah bilangan real, dengan <math>a < b</math>:

: kosong: <math>[b,a] = (a,a) = [a,a) = (a,a] = \{ \} = \emptyset</math>
: degenerasi: <math>[a,a] = \{a\}</math>
: wajar dan berbatas:
:: terbuka: <math>(a,b)=\{x\,|\,a<x<b\}</math>
:: tertutup: <math>[a,b]=\{x\,|\,a\leq x\leq b\}</math>
:: tertutup kiri dan terbuka kanan: <math>[a,b)=\{x\,|\,a\,\leq x<b\}</math>
:: terbuka kiri, tertutup kanan: <math>(a,b]=\{x\,|\,a<x\leq b\}</math>
: berbatas kiri dan tak berbatas kanan:
:: terbuka kiri: <math>(a,\infty)=\{x\,|\,x>a\}</math>
:: tertutup kiri: <math>[a,\infty)=\{x\,|\,x\geq a\}</math>
: tak berbatas kiri dan berbatas kanan:
:: terbuka kanan: <math>(-\infty,b)=\{x\,|\,x<b\}</math>
:: tertutup kanan: <math>(-\infty,b]=\{x\,|\,x\leq b\}</math>
: tak berbatas di kedua ujungnya: <math>(-\infty,+\infty)=\R</math>
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===Intervals of the extended real line===
In some contexts, an interval may be defined as a subset of the [[extended real number line|extended real numbers]], the set of all real numbers augmented with {{math|−∞}} and {{math|+∞}}.

In this interpretation, the notations {{closed-closed|−∞, ''b''}} , {{closed-open|−∞, ''b''}} , {{closed-closed|''a'', +∞}} , and {{open-closed|''a'', +∞}} are all meaningful and distinct.  In particular, {{open-open|−∞, +∞}} denotes the set of all ordinary real numbers, while {{closed-closed|−∞, +∞}} denotes the extended reals.

This choice affects some of the above definitions and terminology. For instance, the interval {{open-open|−∞, +∞}}&nbsp;=&nbsp;<math>\R</math> is closed in the realm of ordinary reals, but not in the realm of the extended reals.

==Properties of intervals==
The intervals are precisely the [[connectedness|connected]] subsets of <math>\R</math>. It follows that the image of an interval by any [[continuous function (topology)|continuous]] function is also an interval. This is one formulation of the [[intermediate value theorem]].

The intervals are also the [[convex set|convex subset]]s of <math>\R</math>.  The interval enclosure of a subset <math>X\subseteq \R</math> is also the [[convex hull]] of <math>X</math>.

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., <math>(a,b) \cup [b,c] = (a,c]</math>).

If <math>\R</math> is viewed as a [[metric space]], its [[open ball]]s are the open bounded sets&nbsp;{{open-open|''c'' + ''r'', ''c'' − ''r''}}, and its [[closed ball]]s are the closed bounded sets&nbsp;{{closed-closed|''c'' + ''r'', ''c'' − ''r''}}.

Any element&nbsp;{{mvar|x}} of an interval&nbsp;{{mvar|I}} defines a partition of&nbsp;{{mvar|I}} into three disjoint intervals {{mvar|I}}<sub>1</sub>, {{mvar|I}}<sub>2</sub>, {{mvar|I}}<sub>3</sub>: respectively, the elements of&nbsp;{{mvar|I}} that are less than&nbsp;{{mvar|x}}, the singleton&nbsp;<math>[x,x] = \{x\}</math>, and the elements that are greater than&nbsp;{{mvar|x}}.  The parts {{mvar|I}}<sub>1</sub> and {{mvar|I}}<sub>3</sub> are both non-empty (and have non-empty interiors) if and only if {{mvar|x}} is in the interior of&nbsp;{{mvar|I}}.  This is an interval version of the [[trichotomy (mathematics)|trichotomy principle]].

==Dyadic intervals==

A ''dyadic interval'' is a bounded real interval whose endpoints are <math>\frac{j}{2^n}</math> and <math>\frac{j+1}{2^n}</math>, where <math>j</math> and <math>n</math> 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]].

Dyadic intervals are relevant to several areas of numerical analysis, including [[adaptive mesh refinement]], [[multigrid methods]] and [[wavelet|wavelet analysis]]. Another way to represent such a structure is [[p-adic analysis]] (for {{math|1=''p'' = 2}}).<ref>{{cite journal |last1=Kozyrev |first1=Sergey |year=2002 |title=Wavelet theory as {{mvar|p}}-adic spectral analysis |journal=[[Izvestiya: Mathematics|Izvestiya RAN. Ser. Mat.]] |volume=66 |issue=2 |pages=149–158 |doi=10.1070/IM2002v066n02ABEH000381 |url=http://mi.mathnet.ru/eng/izv/v66/i2/p149 |accessdate=2012-04-05}}</ref>

== Generalizations ==

=== Multi-dimensional intervals ===
In many contexts, an '''<math>n</math>-dimensional interval''' is defined as a subset of <math>\R^n</math> that is the [[Cartesian product]] of <math>n</math> intervals, <math>I = I_1\times I_2 \times \cdots \times I_n</math>, one on each [[coordinate]] axis.

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

A '''facet''' of such an interval <math>I</math> is the result of replacing any non-degenerate interval factor <math>I_k</math> by a degenerate interval consisting of a finite endpoint of <math>I_k</math>.  The '''faces''' of <math>I</math> comprise <math>I</math> itself and all faces of its facets.  The '''corners''' of <math>I</math> are the faces that consist of a single point of <math>\R^n</math>.

===Complex intervals===
Intervals of [[complex number]]s can be defined as regions of the [[complex plane]], either [[rectangle|rectangular]] or [[disk (mathematics)|circular]].<ref>[http://books.google.com/books?id=Vtqk6WgttzcC Complex interval arithmetic and its applications], Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, ISBN 978-3-527-40134-5</ref>

==Topological algebra==
Intervals can be associated with points of the plane and hence regions of intervals can be associated with [[region (mathematical analysis)|region]]s 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. Often it is assumed that ''y'' > ''x''. For purposes of [[mathematical structure]], this restriction is discarded,<ref>Kaj Madsen (1979) [http://www.ams.org/mathscinet/pdf/586220.pdf Review of "Interval analysis in the extended interval space" by Edgar Kaucher] from [[Mathematical Reviews]]</ref> and "reversed intervals" where ''y'' − ''x'' < 0 are allowed. Then the collection of all intervals [''x,y''] can be identified with the [[topological ring]] formed by the [[direct sum of modules#Direct sum of algebras|direct sum]] of R with itself where addition and multiplication are defined component-wise.

The direct sum algebra <math>( R \oplus R, +, \times)</math> has two [[ideal (ring theory)|ideal]]s, { [''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 [[quadrant (plane geometry)|quadrant]]s 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 <math>R \oplus R</math>, the ring of intervals has been identified<ref>[[D. H. Lehmer]] (1956) [http://www.ams.org/mathscinet/pdf/81372.pdf Review of "Calculus of Approximations"] from Mathematical Reviews</ref> with the [[split-complex number]] plane by M. Warmus and [[D. H. Lehmer]] through the identification
: ''z'' = (''x'' + ''y'')/2 + j (''x'' − ''y'')/2.
This linear mapping of the plane, which amounts of a [[ring isomorphism]], provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as [[polar decomposition#Alternative planar decompositions|polar decomposition]].
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== Lihat pula ==
* [[Deret (matematika)]]
<!--*[[Inequality (mathematics)|Inequality]]
*[[Interval graph]]
*[[Interval finite element]]-->
* [[Urutan]]

== Referensi ==
{{reflist}}

* T. Sunaga, [http://www.cs.utep.edu/interval-comp/sunaga.pdf "Theory of interval algebra and its application to numerical analysis"] {{Webarchive|url=https://web.archive.org/web/20120309164347/http://www.cs.utep.edu/interval-comp/sunaga.pdf |date=2012-03-09 }}, In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp.&nbsp;29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp.&nbsp;126–143.

== Pranala luar ==
* ''A Lucid Interval'' by Brian Hayes: An [http://www.americanscientist.org/issues/pub/a-lucid-interval American Scientist article] provides an introduction.
* [http://id.mind.net/~zona/mmts/miscellaneousMath/intervalNotation/intervalNotation.html Interval Notation Basics] {{Webarchive|url=https://web.archive.org/web/20100208214225/http://id.mind.net/~zona/mmts/miscellaneousMath/intervalNotation/intervalNotation.html |date=2010-02-08 }}
* [http://www.cs.utep.edu/interval-comp/main.html Interval computations website] {{Webarchive|url=https://web.archive.org/web/20060302095039/http://www.cs.utep.edu/interval-comp/main.html |date=2006-03-02 }}
* [http://www.cs.utep.edu/interval-comp/icompwww.html Interval computations research centers] {{Webarchive|url=https://web.archive.org/web/20070203144604/http://www.cs.utep.edu/interval-comp/icompwww.html |date=2007-02-03 }}
* [http://demonstrations.wolfram.com/IntervalNotation/ Interval Notation] by George Beck, [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Interval |urlname=Interval}}

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