Arity

Last updated Nov 1, 2022

The Arity of a Function is the number of arguments it takes. The Arity of a Relation is the number of items it relates.

1. Consider $f: \mathbb{N} \to \mathbb{N}$ defined as $f(x) = x + 1$. This Function has Arity 1. It is said to be Unary.
2. Consider $f: \mathbb{R}^{n} \to \mathbb{R}$ defined as $f(x) = ||\mathbf{x}||_{2}$. This Function has Arity $n \in \mathbb{N}$. It is said to be $n$-ary.
3. Consider the relation $<$ on $\mathbb{R}$. It is $2$-ary because it relates two elements of $\mathbb{R}$ (e.g. $2 < 3$).

Arity is a useful concept for when we talk about Languages, which only consist of Symbols. It doesn’t make as much sense if we are talking about actual mathematical objects because no this depends on interpretation of the input objects. For example, is $x \in \mathbb{R}$ a single input to a function, or is it an infinite number of inputs to a function, when we represent as a Decimal number?