Arity

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

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

${\ displaystyle \ {(a_ {1}, \ dotsc, a_ {k}, b) \ in A_ {1} \ times \ dotsb \ times A_ {k} \ times B | \, f (a_ {1}, \ dotsc, a_ {k}) = b \} \, \ subset \, A_ {1} \ times \ dotsb \ times A_ {k} \ times B}$
passes over, and that is a -digit relation. ${\ displaystyle (k + 1)}$
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}$