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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
Singledigit links only require one argument. An example is the amount function (absolute value) of a number.
Twodigit links take two arguments. Examples of twodigit 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 twodigit 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:
${\ displaystyle A_ {1} = A_ {2} = \ dotsb = A_ {k} = A}$

${\ displaystyle A_ {1} \ times A_ {2} \ times \ dotsb \ times A_ {k} = A ^ {k} = \ {g \ mid g \ colon \, \ {0, \ dotsc, k1 \} \ to A \}}$,
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 zerodigit 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}$${\ displaystyle f \ colon \, \ {0 \} \ to B, \, 0 \ mapsto b_ {0}}$
The algebraic structure of Boolean algebra , which combines all these aspects in itself, can serve as a further example . It has two twodigit operations, union and intersection, the onedigit complement, and two zerodigit operations, the constants and${\ displaystyle (B, \ vee, \ wedge, {} ^ {\ mathrm {C}}, 0,1)}$${\ 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}$${\ displaystyle A}$
A onedigit relation is therefore nothing more than a subset , the zerodigit relations always form the set because of or (empty Cartesian product) . The isomorphism of the relations with predicates assigns the logical (Boolean) constants false (for ) and true (for ) to these two .
${\ displaystyle \ prod _ {i = 1} ^ {0} A_ {i} = \ {\ emptyset \}}$${\ displaystyle A ^ {0} = \ {\ emptyset \}}$${\ displaystyle \ {\ emptyset, \ {\ emptyset \} \}}$${\ displaystyle \ emptyset}$${\ displaystyle \ {\ emptyset \}}$
A typical example of a twodigit relation is

${\ displaystyle \ {(m, m + k) \ mid m, k \ in \ mathbb {N} _ {0} \} \ subset \ mathbb {N} _ {0} \ times \ mathbb {N} _ { 0}}$,
a twodigit relation on the natural numbers , which one usually denotes with . Instead of
writing . Also for any twodigit 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

↑ Empty Cartesian product , is understood as a 0tuple , in connection with strings (words) one also speaks of the empty word , often in characters .${\ displaystyle \ emptyset = ()}$${\ displaystyle \ epsilon}$

↑ instead of conceiving the natural numbers merely as an abstraction that fulfills the Peano axioms .