Lihat sumber Ariti

{{short description|Jumlah tetap argumen atau operan yang diperlukan fungsi atau operasi untuk mendefinisikan hubungan di antara mereka}}

'''Ariti''' ({{IPAc-en|audio=en-us-arity.ogg|ˈ|ær|ᵻ|t|i}}) adalah jumlah [[argumen fungsi|argumen]] atau [[operan]] yang [[fungsi (matematika)|fungsi]] atau [[operasi (matematika)|operasi]] dalam [[logika]], [[matematika]], dan [[ilmu komputer]] membutuhkan. Dalam matematika, ariti juga bisa disebut '''pangkat''',<ref name="Hazewinkel2001">{{cite book|authorlink=Michiel Hazewinkel|first=Michiel|last=Hazewinkel|title=Encyclopaedia of Mathematics, Supplement III|url=https://books.google.com/books?id=47YC2h295JUC&pg=PA3|year=2001|publisher=Springer|isbn=978-1-4020-0198-7|page=3}}</ref><ref name="Schechter1997">{{cite book|first=Eric|last=Schechter|title=Handbook of Analysis and Its Foundations|url=https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA356|year=1997|publisher=Academic Press|isbn=978-0-12-622760-4|page=356}}</ref> tetapi kata ini dapat memiliki banyak arti lain dalam matematika. Dalam logika dan filsafat, itu juga disebut '''adisiti''' dan '''derajat'''.<ref name="DetlefsenBacon1999">{{cite book|first1=Michael |last1=Detlefsen|first2=David Charles|last2=McCarty|first3=John B.|last3=Bacon|title=Logic from A to Z|url=https://archive.org/details/logicfromtoz0000detl|url-access=registration |year=1999|publisher=Routledge|isbn=978-0-415-21375-2|page=[https://archive.org/details/logicfromtoz0000detl/page/7 7]}}</ref><ref name="CocchiarellaFreund2008">{{cite book|first1=Nino B.|last1=Cocchiarella|first2=Max A.|last2=Freund|title=Modal Logic: An Introduction to its Syntax and Semantics|url=https://books.google.com/books?id=zLmxqytfLhgC&pg=PA121|year=2008|publisher=Oxford University Press|isbn=978-0-19-536658-7|page=121}}</ref> Dalam [[linguistik]], biasanya dinamai '''[[valensi (linguistik)|valensi]]'''.<ref name="Crystal2008">{{cite book|first=David|last=Crystal|title=Dictionary of Linguistics and Phonetics|url=https://archive.org/details/dictionarylingui00crys_715|year=2008|publisher=John Wiley & Sons|isbn=978-1-405-15296-9|page=[https://archive.org/details/dictionarylingui00crys_715/page/n532 507]|edition=6th}}</ref>

== Istilah ==
[[Nama Latin]] biasanya digunakan untuk arities tertentu, terutama berdasarkan bahasa Latin [[bilangan distributif]] diartikan "dalam grup ''n''", beberapa [[bilangan kardinal]] atau [[bilangan urutan]] dalam bahasa Latin. Misalnya, 1-ari didasarkan pada kardinal ''tidak digunakan'', bukan dari distributif ''singulī'' yang menghasilkan ''singulari''.

{| class="wikitable"
|-
! '''x-ari''' !! '''Ariti (berbasis Latin)''' !! '''Adiciti (berbasis Yunani)''' !! Contoh dalam matematika !! Contoh dalam ilmu komputer
|-
| 0-ary || ''Nullari'' (dari ''nūllus'') || ''Niladic'' || A [[konstanta (matematika)|konstanta]] || Fungsi tanpa argumen, Benar atau salah
|-
| 1-ari || ''[[Operasi uner|Uner]]'' || ''Monadik'' || [[Invers aditif]] || Operator [[Logika NOT]]
|-
| 2-ary || ''[[Operasi biner|Biner]]'' || ''Diadic'' || [[Penjumlahan]] || ''ATAU'', ''[[eksklusif atau|XOR]]'', ''DAN''
|-
| 3-ari || ''[[Operasi ternari|Ternari]]'' || ''Triadik'' || [[Perkalian tiga|Perkalian tiga vektor]] || [[Operator bersyarat]]
|-
| 4-ari || ''Kuarter'' || ''Tetradik'' || [[Kuartenion]] 
| 
|-
| 5-ari || ''Kuinari''  || ''Pentadik'' || [[Kuantil]] 
| 
|-
| 6-ari || ''Senari'' || ''Heksadik'' ||  || 
|-
| 7-ari || ''Septenari'' || ''Hebdomadik'' ||  || 
|-
| 8-ari || ''Oktonari'' || ''Ogdoadic'' ||  || 
|-
| 9-ari || ''Novenari'' (atau ''nonari'') || ''Enneadik'' ||  || 
|-
| 10-ari || ''Denari'' (atau ''dekenari'') || ''Dekadik'' ||  || 
|-
| Lebih dari 2-ari || ''Multari'' dan ''multiari'' || ''Poliadik'' ||  || 
|-
| Variasai || || ''Variadik'' || Jumlah; misal <math>\sum</math> || [[Fungsi variadik]], [[Pengurangan (pola paralel)|pengurangan]]
|}

''n''-''ari'' berarti operan ''n'' (atau parameter), tetapi digunakan sebagai sinonim dari "poliadik".

Kata ini digunakan untuk mendeskripsikan yang berhubungan dengan bilangan tersebut (misalnya, [[catur undenari]] adalah [[catur varian]] dengan papan 11 × 11, atau [[Petisi Seribu Tahun]] tahun 1603).

Ariti dari [[relasi (matematika)|relasi]] (atau [[predikat (logika matematika)|predikat]]) adalah dimensi dari [[domain fungsi|domain]] dalam korespondensi [[Produk Kartesius]]. Fungsi ariti ''n'' menggunakan ariti ''n'' + 1 disebut sebagai relasi.

== Contoh ==
Istilah "ariti" jarang digunakan dalam penggunaan sehari-hari. Misalnya, dari "ariti dari operasi [[penjumlahan]] adalah 2" atau "penjumlahan adalah operasi dari ariti 2" biasanya disebut "penjumlahan adalah operasi biner". Secara umum, penamaan fungsi atau operasi dengan aritas tertentu menggunakan konvensi yang mirip dengan yang digunakan untuk [[sistem bilangan]] berbasis ''n'' contoh [[Sistem bilangan biner|biner]] dan [[heksadesimal]]. Satu menggabungkan awalan [[Latin]] dengan akhiran ''-ari''; sebagai contoh:

* Fungsi nullari tidak menggunakan argumen.
** Contoh: <math>f()=2</math>
* [[operasi uner|Fungsi uner]] menggunakan satu argumen.
** Contoh: <math>f(x)=2x</math>
* [[operasi biner|Fungsi biner]] menggunakan dua argumen.
** Contoh: <math>f(x,y)=2xy</math>
* [[operasi terner|Fungsi ternari]] menggunakan tiga argumen.
** Contoh: <math>f(x,y,z)=2xyz</math>
* ''n''-fungsi ari diambil dari argumen ''n''.
** Contoh: <math>f(x_1, x_2, \ldots, x_n)=2\prod_{i=1}^n \displaystyle x_i</math>

=== Nullari ===
Kadang-kadang berguna untuk mempertimbangkan [[konstanta (matematika)|konstanta]] sebagai operasi ariti 0, dan karenanya menyebutnya ''nullari''.

Juga, dalam non-[[pemrograman fungsional]], fungsi tanpa argumen dimaknakan dan tidak menggunakan konstanta (karena [[efek samping (ilmu komputer)|efek samping]]). Seringkali, fungsi tersebut sebenarnya memiliki beberapa ''input tersembunyi'' yang termasuk [[variabel global]], termasuk seluruh status sistem (waktu, memori bebas, …). Yang terakhir adalah contoh penting yang digunakan dalam bahasa pemrograman fungsional yaitu "murni".

=== Unari ===
Contoh [[operasi uner]] dalam matematika dan pemrograman termasuk minus dan plus uner, operasi kenaikan dan decrement dalam bahasa gaya [[C (bahasa pemrograman)|C]] (bukan dalam bahasa logika), dan [[Fungsi penerus|penerus]], [[faktorial]], [[invers perkalian|timbal balik]], [[fungsi lantai|lantai]], [[fungsi langit-langit|langit-langit]], [[bagian pecahan]], [[fungsi tanda|tanda]], [[nilai absolut]], [[akar kuadrat]] (akar kuadrat utama), [[konjugat kompleks]] (unari dari bilangan kompleks "satu", yang memiliki dua bagian pada tingkat abstraksi yang lebih rendah), dan fungsi [[Norma (matematika)|norma]] dalam matematika. Operasi [[dua komplemen]], [[Referensi (ilmu komputer)|referensi alamat]] dan [[logika NOT]] adalah contoh operasi unari dalam matematika dan pemrograman.

Semua fungsi dalam [[kalkulus lambda]] dan beberapa [[bahasa pemrograman fungsional]] (terutama yang diturunkan dari [[ML (bahasa pemrograman)|ML]]) secara teknis unary, tetapi lihat [[#n-ari|n-ari]] di bawah.

Menurut [[Willard Van Orman Quine|Quine]], ''distributives'' dalam bahasa Latin menjadi ''singuli, bini, terni,'' dan seterusnya, istilah "singulari" adalah kata sifat yang benar, bukan "unari."<ref>{{Citation
  | last = Quine
  | first = W. V. O.
  | title = Mathematical logic
  | year = 1940
  | place = Cambridge, Massachusetts
  | publisher = Harvard University Press
  | page=13
}}</ref> [[Abraham Robinson]] mengikuti penggunaan Quine.<ref>{{Citation
  | last = Robinson
  | first = Abraham
  | title = "Non-standard Analysis"
  | year = 1966
  | place = Amsterdam
  | publisher = North-Holland
  | page=19
}}</ref>
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===Binary===
Most operators encountered in programming and mathematics are of the [[binary operation|binary]] form. For both programming and mathematics, these can be the [[multiplication operator]], the radix operator, the often omitted [[exponentiation]] operator, the [[logarithm]] operator, the addition operator, the division operator. Logical predicates such as ''OR'', ''[[exclusive or|XOR]]'', ''AND'', ''IMP'' are typically used as binary operators with two distinct operands. In [[Complex instruction set computing|CISC]] architectures, it is common to have two source operands (and store result in one of them).

===Ternary===
Common ternary operations besides generic function in mathematics are the [[summatory]] and the [[productory]] though some other n-ary operation may be implied.

The computer programming language [[C (programming language)|C]] and its various descendants (including [[C++]], [[C Sharp (programming language)|C#]], [[Java (programming language)|Java]], [[Julia (programming language)|Julia]], [[Perl]], and others) provides the [[ternary operator]] <code>[[?:]]</code>, also known as the [[conditional operator]], taking three operands. The first operand (the condition) is evaluated, and if it is true, the result of the entire expression is the value of the second operand, otherwise it is the value of the third operand. The [[Forth (programming language)|Forth]] language also contains a ternary operator, <code>*/</code>, which multiplies the first two (one-cell) numbers, dividing by the third, with the intermediate result being a double cell number. This is used when the intermediate result would overflow a single cell. The [[Python (programming language)|Python]] language has a ternary conditional expression, <code>x if C else y</code>. The Unix [[dc (computer program)|dc calculator]] has several ternary operators, such as <code>|</code>, which will pop three values from the stack and efficiently compute <math display="inline">x^y \bmod z</math> with [[arbitrary-precision arithmetic|arbitrary precision]]. Additionally, many ([[Reduced instruction set computing|RISC]]) [[assembly language]] instructions are ternary (as opposed to only two operands specified in CISC); or higher, such as <syntaxhighlight lang="asm" inline>MOV %AX, (%BX, %CX)</syntaxhighlight>, which will load (MOV) into register {{mono|AX}} the contents of a calculated memory location that is the sum (parenthesis) of the registers {{mono|BX}} and {{mono|CX}}.-->
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===''n''-ary===
From a mathematical point of view, a function of ''n'' arguments can always be considered as a function of one single argument which is an element of some [[product space]]. However, it may be convenient for notation to consider ''n''-ary functions, as for example [[multilinear map]]s (which are not linear maps on the product space, if {{nowrap|''n'' ≠ 1}}).

The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some [[object composition|composite type]] such as a [[tuple]], or in languages with [[higher-order function]]s, by [[currying]].

===Varying arity===
In computer science, a function accepting a variable number of arguments is called ''[[variadic function|variadic]]''. In logic and philosophy, predicates or relations accepting a variable number of arguments are called ''[[multigrade predicate|multigrade]]'', anadic, or variably polyadic.<ref>{{Cite journal | doi = 10.1093/mind/113.452.609 | last1 = Oliver | first1 = Alex | year = 2004 | title = Multigrade Predicates | url = | journal = Mind | volume = 113 | issue = | pages = 609–681 }}</ref>-->

== Lihat pula ==
{{Portal|Matematika|Filsafat}}
{{Div col|colwidth=30em}}
* [[Logika kerabat]]
* [[Hubungan biner]]
* [[Hubungan terner]]
* [[Teori hubungan]]
* [[Tanda tangan (logika)]]
* [[Parameter]]
* [[Bilangan p-adic|'' p '' - bilangan adic]]
* [[Kardinalitas]]
* [[Valensi (linguistik)|Valensi]]
* [[kode n-ari|'' n '' - kode ary]]
* [[grup n-ari|'' n '' - grup ary]]
{{colend}}

== Referensi ==
{{Reflist}}

== Pranala luar ==
{{wiktionary|Appendix:English arities and adicities}}
Sebuah monograf tersedia online gratis:

* Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]''  Springer-Verlag. {{ISBN|3-540-90578-2}}. Especially pp.&nbsp;22–24.

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