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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:
For example is a two-digit shortcut.
The following applies in particular:
Also can because of
a zero-digit link always as a constant mapping
be considered. This mapping can in turn be understood as the constant .
For example, simple can also be used for the link .
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 .
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
Arity of relations
More generally, a subset is called a -digit relation. Is , then one speaks of a -digit relation on .
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 .
A typical example of a two-digit relation is
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 .
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
to your function graph
passes over, and that is a -digit relation.
↑ Empty Cartesian product , is understood as a 0-tuple , in connection with strings (words) one also speaks of the empty word , often in characters .
↑ instead of conceiving the natural numbers merely as an abstraction that fulfills the Peano axioms .