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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:
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e02832d21ef514a2e224987dd3be1ca81f06bc7)
For example is a two-digit shortcut.
![f \ colon \, \ R \ times \ N \ to \ R, \, (x, n) \ mapsto f (x, n): = x ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f25226d6ee4901509a3c199e9f6a558654fa80)
The following applies in particular:
![{\ displaystyle A_ {1} = A_ {2} = \ dotsb = A_ {k} = A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5b38ac9b4708895666bfbdb310f2967271d10a)
-
,
so then
-
.
Also can because of
![{\ displaystyle A ^ {0} = \ {g \ mid g \ colon \, \ emptyset \ to A \} = \ {\ emptyset \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47c27309312973ef5d13d3a0b0d8fb9309bf892e)
a zero-digit link always as a constant mapping
![{\ displaystyle f \ colon \, \ {\ emptyset \} \ to B, \, \ emptyset \ mapsto b_ {0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5905a4043c0904991f84c830616f584236c1507)
be considered. This mapping can in turn be understood as the constant .
![f \ in B ^ 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a0acadbac2b1d5a4af9b35ebeead5681f4779c)
![b_0 \ in B](https://wikimedia.org/api/rest_v1/media/math/render/svg/576f1554c09ba87c0bfcd2e0904d9b8228552808)
For example, simple can also be used for the link .
![{\ displaystyle f \ colon \, \ mathbb {N} ^ {0} \ to \ mathbb {N}, \, \ emptyset \ mapsto 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d4fa5280cf85cf449ab5948a54838369e4bb48)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1719298cfa96cf541393c5e3b511a8c2b59e27)
![A ^ {0} = \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16c7a5fadaaf442e55febcb457dfc91143b3aafb)
![b_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e425056f502ca07b103ffbf6ac4720e0f8a01f0)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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![(B, \ vee, \ wedge, {} ^ {\ mathrm C}, 0, 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7479a14ab249270a65726396ce9bad9e14b633)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99af9d7fda305d526f0e2fd8c80d8a4d857154ea)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle A_ {1} = \ dotsb = A_ {k} = A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c451f95552ca56cd23d26ba56c1c4e4833a412a2)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
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 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/311476ddf3c2183fe080d47fbaf24d9bc38dd433)
![A ^ 0 = \ {\ emptyset \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3084ae1df12463e5c258bbb7efd5388db694fea)
![{\ displaystyle \ {\ emptyset, \ {\ emptyset \} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83f67b04150799fb0d9f646e1b5965fa5cf5b32)
![\ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7)
![\ {\ emptyset \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2943085c6db53add570975fb7c528c398f0c52)
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 .
![\ mathbb {N} _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab7e98123f0def29a1cd3df96a0b7a58f4202c)
![\ leq](https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035)
![{\ displaystyle (m, n) \ in {\ leq}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/499703886e90b75a56775a72e622c412985c0b8f)
![m \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/0017737947454c2911336b2d038c91f5a7a70bc0)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![(x, y) \ in R](https://wikimedia.org/api/rest_v1/media/math/render/svg/564266a1c3efe90b1974df60a445161fdf58f14e)
![xRy](https://wikimedia.org/api/rest_v1/media/math/render/svg/324aab4e2674bb19cc073ea887888b98f0fc63d4)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9ad74f30d22787748abbf5996a0b114e0752c4)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0f08e7ad0345b44ce79fbbdb79a47f26fa7c09)
passes over, and that is a -digit relation.
![(k + 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9f13644a6be482d7ddb19a6e0c706564773085)
Remarks
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↑ 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 = ()}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e65fc1db0b974a4c9dec109869f80e6f065fd266)
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↑ instead of conceiving the natural numbers merely as an abstraction that fulfills the Peano axioms .