Editing Finitary relation (section)

== Definitions ==
{{Blockquote|When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation.|[[Augustus De Morgan]]{{sfn|ps=|De Morgan|1966}}}}

; Definition : ''R'' is an ''n''-ary '''relation''' on sets {{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}} is given by a subset of the Cartesian product {{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}.{{sfn|ps=|Codd|1970}}

Since the definition is predicated on the underlying sets {{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}}, ''R'' may be more formally defined as the ({{nowrap|''n'' + 1}})-tuple {{nowrap|(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, ''G'')}}, where ''G'', called the ''graph'' of ''R'', is a subset of the Cartesian product {{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}.

As is often done in mathematics, the same symbol is used to refer to the mathematical object and an underlying set, so the statement {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∈ ''R''}} is often used to mean {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∈ ''G''}} is read "''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> are ''R''-related" and are denoted using [[Polish notation|prefix notation]] by {{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''n''</sub>}} and using [[Reverse Polish notation|postfix notation]] by {{nowrap|''x''<sub>1</sub>⋯''x''<sub>''n''</sub>''R''}}. In the case where ''R'' is a binary relation, those statements are also denoted using [[infix notation]] by {{nowrap|''x''<sub>1</sub>''Rx''<sub>2</sub>}}.

The following considerations apply:
* The set ''X''<sub>''i''</sub> is called the {{itco|{{mvar|i}}}}th ''domain'' of ''R''.{{sfn|ps=|Codd|1970}} In the case where ''R'' is a binary relation, ''X''<sub>1</sub> is also called simply the [[Binary relation#Definition|''domain'']] or ''set of departure'' of ''R'', and ''X''<sub>2</sub> is also called the [[Binary relation#Definition|''codomain'']] or ''set of destination'' of ''R''.
* When the elements of ''X''<sub>''i''</sub> are relations, ''X''<sub>''i''</sub> is called a ''nonsimple domain'' of ''R''.{{sfn|ps=|Codd|1970}}
* The set of {{nowrap|∀''x''<sub>''i''</sub> ∈ ''X''<sub>''i''</sub>}} such that {{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''i''−1</sub>''x''<sub>''i''</sub>''x''<sub>''i''+1</sub>⋯''x''<sub>''n''</sub>}} for at least one {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} is called the ''i''th ''domain of definition'' or ''active domain'' of ''R''.{{sfn|ps=|Codd|1970}} In the case where ''R'' is a binary relation, its first domain of definition is also called simply the [[Binary relation#Definition|''domain of definition'']] or ''active domain'' of ''R'', and its second domain of definition is also called the [[Binary relation#Definition|''codomain of definition'']] or ''active codomain'' of ''R''.
* When the {{mvar|i}}th domain of definition of ''R'' is equal to ''X''<sub>''i''</sub>, ''R'' is said to be ''total'' on its ''i''th domain (or on ''X''<sub>''i''</sub>, when this is not ambiguous). In the case where ''R'' is a binary relation, when ''R'' is total on ''X''<sub>1</sub>, it is also said to be [[Binary relation#Special types of binary relations|''left-total'']] or ''serial'', and when ''R'' is total on ''X''<sub>2</sub>, it is also said to be [[Binary relation#Special types of binary relations|''right-total'']] or ''surjective''.
* When {{nowrap|∀''x'' ∀''y'' ∈ ''X''<sub>''i''</sub>.}} {{nowrap|∀''z'' ∈ ''X''<sub>''j''</sub>.}} {{nowrap|1=''xR''<sub>''ij''</sub>''z'' &and; ''yR''<sub>''ij''</sub>''z'' ⇒ ''x'' = ''y''}}, where {{nowrap|''i'' ∈ ''I''}}, {{nowrap|''j'' ∈ ''J''}}, {{nowrap|1=''R''<sub>''ij''</sub> = ''π''<sub>''ij''</sub> ''R''}}, and {{nowrap|{{mset|''I'', ''J''}}}} is a [[Partition of a set|partition]] of {{nowrap|{{mset|1, ..., ''n''}}}}, ''R'' is said to be ''unique'' on {{nowrap|{{mset|''X''<sub>''i''</sub>}}<sub>''i''∈''I''</sub>}}, and {{nowrap|{{mset|''X''<sub>''i''</sub>}}<sub>''i''∈''J''</sub>}} is called ''a [[primary key]]''{{sfn|ps=|Codd|1970}} of ''R''. In the case where ''R'' is a binary relation, when ''R'' is unique on {{mset|''X''<sub>1</sub>}}, it is also said to be [[Binary relation#Special types of binary relations|''left-unique'']] or ''injective'', and when ''R'' is unique on {{mset|''X''<sub>2</sub>}}, it is also said to be [[Binary relation#Special types of binary relations|''univalent'']] or ''right-unique''.
* When all ''X''<sub>''i''</sub> are the same set ''X'', it is simpler to refer to ''R'' as an ''n''-ary relation over ''X'', called a ''[[homogeneous relation]]''. Without this restriction, ''R'' is called a ''[[heterogeneous relation]]''.
* When any of ''X''<sub>''i''</sub> is empty, the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation {{nowrap|1=''R'' = ∅}}.

Let a [[Boolean domain]] ''B'' be a two-element set, say, {{nowrap|1=''B'' = {{mset|0, 1}}}}, whose elements can be interpreted as logical values, typically {{nowrap|1=0 = false}} and {{nowrap|1=1 = true}}. The [[Indicator function|characteristic function]] of ''R'', denoted by ''χ''<sub>''R''</sub>, is the [[Boolean-valued function]] {{nowrap|''χ''<sub>''R''</sub>: ''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub> → ''B''}}, defined by {{nowrap|1=''χ''<sub>''R''</sub>({{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}) = 1}} if {{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''n''</sub>}} and {{nowrap|1=''χ''<sub>''R''</sub>({{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}) = 0}} otherwise.

In applied mathematics, [[computer science]] and statistics, it is common to refer to a Boolean-valued function as an ''n''-ary [[Predicate (mathematics)|''predicate'']]. From the more abstract viewpoint of [[formal logic]] and [[model theory]], the relation ''R'' constitutes a ''logical model'' or a ''relational structure'', that serves as one of many possible [[Interpretation (logic)|interpretation]]s of some ''n''-ary predicate symbol.

Because relations arise in many scientific disciplines, as well as in many branches of [[mathematics]] and [[logic]], there is considerable variation in terminology. Aside from the [[Set theory|set-theoretic]] [[extension (semantics)|extension]] of a relational concept or term, the term "relation" can also be used to refer to the corresponding logical entity, either the [[Comprehension (logic)|logical comprehension]], which is the totality of [[intension]]s or abstract properties shared by all elements in the relation, or else the symbols denoting these elements and intensions. Further, some writers of the latter persuasion introduce terms with more concrete connotations (such as "relational structure" for the set-theoretic extension of a given relational concept).