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←Membuat halaman berisi '<!--{{About|intervals of real numbers and other totally ordered sets|the most general definition|partially ordered set|other uses|Interval (disambiguation)}}--> '''In...'
 
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==Notasi bagi interval==
Interval angka-angka antara {{mvar|a}} dan {{mvar|b}}, termasuk {{mvar|a}} dan {{mvar|b}}, sering dilambangkan dengan {{closedtutup-closedtutup|''a'', ''b''}}. Dua bilangan itu disebut "titik-titik ujung" (''endpoints'') suatu interval. Pada 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 ===
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[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R\,|\,a\le x\le b\}.
\end{align} </math>
Perhatikan bahwa {{openbuka-openbuka|''a'', ''a''}}, {{closedtutup-openbuka|''a'', ''a''}}, dan {{openbuka-closedtutup|''a'', ''a''}} melambangkan [[himpunan kosong]], sedangkan {{closedtutup-closedtutup|''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]].
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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}}.
 
==TerminologyTerminologi==
'''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}}.
An '''open interval''' does not include its endpoints, and is indicated with parentheses. For example {{open-open|0,1}} means greater than {{math|0}} and less than {{math|1}}. A '''closed interval''' includes its endpoints, and is denoted with square brackets. For example {{closed-closed|0,1}} means greater than or equal to {{math|0}} and less than or equal to {{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}}.
A '''degenerate interval''' is any [[singleton set|set consisting of a single real number]]. 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.
 
'''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..
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'''.
 
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.
 
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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 interval==
Interval 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>
==Classification of intervals==
:: closeddegenerasi: <math>[a,ba] = \{x\,|\,a\leq x\leq b\}</math>▼
The intervals of real numbers can be classified into eleven different types, listed below; where {{mvar|a}} and {{mvar|b}} are real numbers, with <math>a < b</math>:
: proper anddan boundedberbatas:▼
 
:: emptyterbuka: <math>[b,a] = (a,ab) = [a\{x\,a) = (a|\,a] = \{ <x<b\} = \emptyset</math>
:: degeneratetertutup: <math>[a,ab] = \{x\,|\,a\leq x\leq b\}</math>
:: right-closedtertutup kiri dan terbuka kanan: <math>(-\infty[a,b])=\{x\,|\,xa\,\leq x<b\}</math>▼
▲: proper and bounded:
:: openterbuka kiri, tertutup kanan: <math>(a,b)]=\{x\,|\,a<x<\leq b\}</math>
: berbatas kiri dan tak berbatas kanan:
▲:: closed: <math>[a,b]=\{x\,|\,a\leq x\leq b\}</math>
:: left-closed,terbuka right-openkiri: <math>[(a,b\infty)=\{x\,|\,x>a\,\leq x<b\}</math>
:: left-open,tertutup right-closedkiri: <math>([a,b]\infty)=\{x\,|\,a<x\leqgeq ba\}</math>
: tak berbatas kiri dan berbatas kanan:
: left-bounded and right-unbounded:
:: left-openterbuka kanan: <math>(a,-\infty,b)=\{x\,|\,x>a<b\}</math>
:: left-closedtertutup kanan: <math>[a,(-\infty),b]=\{x\,|\,x\geqleq ab\}</math>
: unboundedtak atberbatas bothdi endskedua ujungnya: <math>(-\infty,+\infty)=\R</math>▼
: left-unbounded and right-bounded:
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:: right-open: <math>(-\infty,b)=\{x\,|\,x<b\}</math>
▲:: right-closed: <math>(-\infty,b]=\{x\,|\,x\leq b\}</math>
▲: unbounded at both ends: <math>(-\infty,+\infty)=\R</math>
 
===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|+∞}}.
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