Arity () is the number of
arguments
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
or
operand
In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Example
The following arithmetic expression shows an example of operators and operands:
:3 + 6 = 9
In the above examp ...
s taken by a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
,
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
or
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. In mathematics, arity may also be named ''rank'',
but this word can have many other meanings in mathematics. In logic and
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
, it is also called adicity and degree.
In
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
, it is usually named
valency.
Examples
The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
s such as
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
and
hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
. One combines a
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
prefix with the -ary ending; for example:
* A nullary function takes no arguments.
** Example:
* A
unary function
A unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range.
Examples ...
takes one argument.
** Example:
* A
binary function
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.
Precisely stated, a function f is binary if there exists sets X, Y, Z such that
:\,f \colon X \times Y \rightar ...
takes two arguments.
** Example:
* A
ternary function takes three arguments.
** Example:
* An ''n''-ary function takes ''n'' arguments.
** Example:
Nullary
Sometimes it is useful to consider a
constant to be an operation of arity 0, and hence call it ''nullary''.
Also, in non-
functional programming
In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declar ...
, a function without arguments can be meaningful and not necessarily constant (due to
side effect
In medicine, a side effect is an effect, whether therapeutic or adverse, that is secondary to the one intended; although the term is predominantly employed to describe adverse effects, it can also apply to beneficial, but unintended, consequence ...
s). Often, such functions have in fact some ''hidden input'' which might be
global variable
In computer programming, a global variable is a variable with global scope, meaning that it is visible (hence accessible) throughout the program, unless shadowed. The set of all global variables is known as the ''global environment'' or ''global s ...
s, including the whole state of the system (time, free memory, ...). The latter are important examples which usually also exist in "purely" functional programming languages.
Unary
Examples of
unary operator
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on ...
s in mathematics and in programming include the unary minus and plus, the increment and decrement operators in
C-style languages (not in logical languages), and the
successor
Successor may refer to:
* An entity that comes after another (see Succession (disambiguation))
Film and TV
* ''The Successor'' (film), a 1996 film including Laura Girling
* ''The Successor'' (TV program), a 2007 Israeli television program Mus ...
,
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
,
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
,
floor
A floor is the bottom surface of a room or vehicle. Floors vary from simple dirt in a cave to many layered surfaces made with modern technology. Floors may be stone, wood, bamboo, metal or any other material that can support the expected load ...
,
ceiling
A ceiling is an overhead interior surface that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings ...
,
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
,
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
,
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
,
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
(the principal square root),
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
(unary of "one" complex number, that however has two parts at a lower level of abstraction), and
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
functions in mathematics. The
two's complement
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
,
address reference and the
logical NOT
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
operators are examples of unary operators in math and programming.
All functions in
lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation ...
and in some
functional programming language
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that m ...
s (especially those descended from
ML) are technically unary, but see
n-ary
-ary may refer to:
* The arity of a function, operation, or relation
** -ary associativity, a specific rule attached to -ary functions
*** -ary group, a generalization of group
* The radix of a numerical representation system
* The number of le ...
below.
According to
Quine, the Latin distributives being ''singuli, bini, terni,'' and so forth, the term "singulary" is the correct adjective, rather than "unary."
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
follows Quine's usage.
In philosophy, the adjective ''monadic'' is sometimes used to describe a
one-place relation such as 'is square-shaped' as opposed to a
two-place relation such as 'is the sister of'.
Binary
Most operators encountered in programming and mathematics are of the
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
form. For both programming and mathematics, these include the
multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
, the radix operator, the often omitted
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
operator, the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
operator, the
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
operator, and the
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
operator. Logical predicates such as ''
OR'', ''
XOR
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
'', ''
AND
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boole ...
'', ''IMP'' are typically used as binary operators with two distinct operands. In
CISC architectures, it is common to have two source operands (and store result in one of them).
Ternary
The computer programming language
C and its various descendants (including
C++
C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
,
C#,
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
,
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g. ...
,
Perl
Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...
, and others) provide the
ternary conditional operator
In computer programming, the ternary conditional operator is a ternary operator that is part of the syntax for basic conditional expressions in several programming languages. It is commonly referred to as the conditional operator, ternary if, or ...
?:
. 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
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
language has a ternary conditional expression,
x if C else y
.
The
Forth
Forth or FORTH may refer to:
Arts and entertainment
* ''forth'' magazine, an Internet magazine
* ''Forth'' (album), by The Verve, 2008
* ''Forth'', a 2011 album by Proto-Kaw
* Radio Forth, a group of independent local radio stations in Scotla ...
language also contains a ternary operator,
*/
, 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 Unix
dc calculator has several ternary operators, such as
,
, which will pop three values from the stack and efficiently compute
with
arbitrary precision
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are lim ...
.
Many (
RISC
In computer engineering, a reduced instruction set computer (RISC) is a computer designed to simplify the individual instructions given to the computer to accomplish tasks. Compared to the instructions given to a complex instruction set comput ...
)
assembly language
In computer programming, assembly language (or assembler language, or symbolic machine code), often referred to simply as Assembly and commonly abbreviated as ASM or asm, is any low-level programming language with a very strong correspondence be ...
instructions are ternary (as opposed to only two operands specified in CISC); or higher, such as
MOV %AX, (%BX, %CX), which will load (MOV) into register the contents of a calculated memory location that is the sum (parenthesis) of the registers and .
''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
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. However, it may be convenient for notation to consider ''n''-ary functions, as for example
multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W are ...
s (which are not linear maps on the product space, if ).
The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some
composite type
In computer science, a composite data type or compound data type is any data type which can be constructed in a program using the programming language's primitive data types and other composite types. It is sometimes called a structure or aggrega ...
such as a
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
, or in languages with
higher-order function
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following:
* takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself ...
s, by
currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f that ...
.
Varying arity
In computer science, a function accepting a variable number of arguments is called ''
variadic In computer science, an operator or function is variadic if it can take a varying number of arguments; that is, if its arity is not fixed.
For specific articles, see:
* Variadic function
* Variadic macro in the C preprocessor
* Variadic template
...
''. In logic and philosophy, predicates or relations accepting a variable number of arguments are called ''
multigrade'', anadic, or variably polyadic.
Terminology
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
ate names are commonly used for specific arities, primarily based on Latin
distributive number
In linguistics, a distributive numeral, or distributive number word, is a word that answers "how many times each?" or "how many at a time?", such as ''singly'' or ''doubly''. They are contrasted with multipliers. In English, this part of speec ...
s meaning "in group of ''n''", though some are based on Latin
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s or
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s. For example, 1-ary is based on cardinal ''unus'', rather than from distributive ''singulī'' that would result in ''singulary''.
''n''-''ary'' means ''n'' operands (or parameters), but is often used as a synonym of "polyadic".
These words are often used to describe anything related to that number (e.g., undenary chess is a
chess variant
A chess variant is a game related to, derived from, or inspired by chess. Such variants can differ from chess in many different ways.
"International" or "Western" chess itself is one of a family of games which have related origins and could be co ...
with an 11×11 board, or the
Millenary Petition
The Millenary Petition was a list of requests given to James I by Puritans in 1603 when he was travelling to London in order to claim the English throne. It is claimed, but not proven, that this petition had 1,000 signatures of Puritan ministers ...
of 1603).
The arity of a
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
(or
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
) is the dimension of the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
in the corresponding
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
. (A function of arity ''n'' thus has arity ''n''+1 considered as a relation.)
In
computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, there is often a
syntactical
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
distinction between
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
and
functions; syntactical operators usually have arity 0, 1, or 2 (the
ternary operator
In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''.
In computer science, a ternary operator ...
?: is also common). Functions vary widely in the number of arguments, though large numbers can become unwieldy. Some programming languages also offer support for
variadic functions
In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages.
The term ''va ...
, i.e., functions syntactically accepting a variable number of arguments.
See also
*
Logic of relatives
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
*
Binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
*
Ternary relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.
Just as a binary relation ...
*
Theory of relations
In mathematics, a finitary relation over sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples consisting of elements ''x'i'' in ''X'i''. Typically, the relation describes a possible connection between the elem ...
*
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are ...
*
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
*
''p''-adic number
*
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
*
Valency
*
''n''-ary code
*
''n''-ary group
*
*
References
External links
A monograph available free online:
* Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981.
A Course in Universal Algebra.' Springer-Verlag. . Especially pp. 22–24.
{{Mathematical logic
Abstract algebra
Universal algebra
cs:Operace (matematika)#Arita operace