Arity

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The term arity (also arity ; English arity ) represents the number of arguments of a link , an image or an operator or the computer science for the number of parameters of functions , procedures or methods . More generally, this term can also be applied to relations.

Arity for illustrations

Single-digit links only require one argument. An example is the amount function (absolute value) of a number.

Two-digit links take two arguments. Examples of two-digit links are the arithmetic operations addition , subtraction , multiplication , or division , or the logical operations and (logical) , or or (logical) .

A -digit link ,, is therefore a mapping with arguments: ${\displaystyle k}$ ${\displaystyle k>0}$ ${\displaystyle k}$

${\displaystyle f\colon \,A_{1}\times A_{2}\times \dotsb \times A_{k}\to B,\,(a_{1},\dotsc ,a_{k})\mapsto f(a_{1},\dotsc ,a_{k}).}$

For example is a two-digit shortcut. ${\displaystyle f\colon \,\mathbb {R} \times \mathbb {N} \to \mathbb {R} ,\,(x,n)\mapsto f(x,n):=x^{n}}$

The following applies in particular: ${\displaystyle A_{1}=A_{2}=\dotsb =A_{k}=A}$

${\displaystyle A_{1}\times A_{2}\times \dotsb \times A_{k}=A^{k}=\{g\mid g\colon \,\{0,\dotsc ,k-1\}\to A\}}$,

so then

${\displaystyle f\colon \,A^{k}\to B}$.

Also can because of

${\displaystyle A^{0}=\{g\mid g\colon \,\emptyset \to A\}=\{\emptyset \}}$

a zero-digit link always as a constant mapping

${\displaystyle f\colon \,\{\emptyset \}\to B,\,\emptyset \mapsto b_{0},}$

be considered. This mapping can in turn be understood as the constant . ${\displaystyle f\in B^{1}}$ ${\displaystyle b_{0}\in B}$

For example, simple can also be used for the link . ${\displaystyle f\colon \,\mathbb {N} ^{0}\to \mathbb {N} ,\,\emptyset \mapsto 1,}$ ${\displaystyle 1}$

If the set theoretic representation according to John von Neumann is used as a basis for the natural numbers , then is and thus . For a constant in then as picture construed . ${\displaystyle 0=\emptyset ,1=\{\emptyset \},\dots }$ ${\displaystyle A^{0}=\{0\}}$ ${\displaystyle b_{0}}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle f\colon \,\{0\}\to B,\,0\mapsto b_{0}}$

The algebraic structure of Boolean algebra , which combines all these aspects in itself, can serve as a further example . It has two two-digit operations, union and intersection, the one-digit complement, and two zero-digit operations, the constants and ${\displaystyle (B,\vee ,\wedge ,{}^{\mathrm {C} },0,1)}$ ${\displaystyle 0}$ ${\displaystyle 1.}$

Arity of relations

More generally, a subset is called a -digit relation. Is , then one speaks of a -digit relation on . ${\displaystyle R\subset A_{1}\times A_{2}\times \dotsb \times A_{k}}$ ${\displaystyle k}$ ${\displaystyle A_{1}=\dotsb =A_{k}=A}$ ${\displaystyle k}$ ${\displaystyle A}$

A one-digit relation is therefore nothing more than a subset , the zero-digit relations always form the set because of or (empty Cartesian product) . The isomorphism of the relations with predicates assigns the logical (Boolean) constants false (for ) and true (for ) to these two . ${\displaystyle \prod _{i=1}^{0}A_{i}=\{\emptyset \}}$ ${\displaystyle A^{0}=\{\emptyset \}}$ ${\displaystyle \{\emptyset ,\{\emptyset \}\}}$ ${\displaystyle \emptyset }$ ${\displaystyle \{\emptyset \}}$

A typical example of a two-digit relation is

${\displaystyle \{(m,m+k)\mid m,k\in \mathbb {N} _{0}\}\subset \mathbb {N} _{0}\times \mathbb {N} _{0}}$,

a two-digit relation on the natural numbers , which one usually denotes with . Instead of writing . Also for any two-digit relations we like to reproduce as for better readability . ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle \leq }$ ${\displaystyle (m,n)\in {\leq }}$ ${\displaystyle m\leq n}$ ${\displaystyle R}$ ${\displaystyle (x,y)\in R}$ ${\displaystyle xRy}$

If one considers that mappings are special relations, the definitions of arity given here for mappings and relations do not coincide. If one treats a function as a relation, this means that one of the function

${\displaystyle f\colon \,A_{1}\times \dotsb \times A_{k}\to B}$

to your function graph

${\displaystyle \{(a_{1},\dotsc ,a_{k},b)\in A_{1}\times \dotsb \times A_{k}\times B|\,f(a_{1},\dotsc ,a_{k})=b\}\,\subset \,A_{1}\times \dotsb \times A_{k}\times B}$

passes over, and that is a -digit relation. ${\displaystyle (k+1)}$

Remarks

1. ↑ Empty Cartesian product , is understood as a 0-tuple , in connection with strings (words) one also speaks of the empty word , often in characters . ${\displaystyle \emptyset =()}$ ${\displaystyle \epsilon }$
2. ↑ instead of conceiving the natural numbers merely as an abstraction that fulfills the Peano axioms .