Editing Finitary relation (section)

{{short description|Property that assigns truth values to k-tuples of individuals}}

In [[mathematics]], a '''finitary relation''' over a sequence of sets {{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}} is a [[subset]] of the [[Cartesian product]] {{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}; that is, it is a set of ''n''-tuples {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}, each being a sequence of elements ''x''<sub>''i''</sub> in the corresponding ''X''<sub>''i''</sub>.{{sfn|ps=|Codd|1970}}<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Relation|title=Relation – Encyclopedia of Mathematics|website=www.encyclopediaofmath.org|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.cs.odu.edu/~toida/nerzic/content/relation/definition/cp_gen/index.html|title=Definition of ''n''-ary Relation|website=cs.odu.edu|access-date=2019-12-12}}</ref> Typically, the relation describes a possible connection between the elements of an ''n''-tuple. For example, the relation "''x'' is divisible by ''y'' and ''z''" consists of the set of 3-tuples such that when substituted to ''x'', ''y'' and ''z'', respectively, make the sentence true.

The non-negative integer ''n'' that gives the number of "places" in the relation is called the ''[[arity]]'', ''adicity'' or ''degree'' of the relation. A relation with ''n'' "places" is variously called an '''''n''-ary relation''', an '''''n''-adic relation''' or a '''relation of degree ''n'''''. Relations with a finite number of places are called ''finitary relations'' (or simply ''relations'' if the context is clear). It is also possible to generalize the concept to ''infinitary relations'' with [[Sequence|infinite sequences]].{{sfn|ps=|Nivat|1981}}