Arity

Definition

Let $A$ and $B$ be sets and let $n \ge 0$ be an integer. Write $A^n$ for the Cartesian product of $n$ copies of $A$ , the set of $n$ -tuples of elements of $A$ . Here $A^0$ is the set whose only element is the empty tuple.

A function $f \colon A^n \to B$ has arity $n$ .
A relation $R \subseteq A^n$ has arity $n$ .

A function or relation of arity $n$ is called $n$ -ary.

Remarks

Example: the map $(x, y, z) \mapsto x y + z$ from $\R^3$ to $\R$ has arity $3$ , because its domain is $\R^3$ .

Non-example: the map $\R \to \R^2$ , $x \mapsto (x, x)$ , has arity $1$ and not $2$ , because the arity counts the factors of the domain, not of the codomain.

The arity of a function depends on how its domain is written as a power. A map on $\R^2$ can be seen as a binary function on $\R$ or as a unary function on the set $\R^2$ . The words nullary, unary, binary and ternary are used for arities $0$ , $1$ , $2$ and $3$ .

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Cartesian product concept
Function concept
Arity definition

Tag	`YZXPWC2MT1`
URI	`m2://YZXPWC2MT1`
Kind	definition
Writing	W0 stub
MSC	08A02, 03C07
Defines	Arity
Uses	Function, Cartesian product
Changed	2 November 2026

Arity

Definition

Remarks

Reading path

Further reading

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