Relation Function

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A relation function is a formal function whose function range is a binary set.

References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/finitary_relation Retrieved:2016-1-18.
• In mathematics, a finitary relation has a finite number of "places". In set theory and logic, a relation is a property that assigns truth values to $\displaystyle{ k }$ -tuples of individuals. Typically, the property describes a possible connection between the components of a $\displaystyle{ k }$ -tuple. For a given set of $\displaystyle{ k }$ -tuples, a truth value is assigned to each $\displaystyle{ k }$ -tuple according to whether the property does or does not hold.

An example of a ternary relation (i.e., between three individuals) is: " $\displaystyle{ X }$ was introduced to $\displaystyle{ Y }$ by $\displaystyle{ Z }$ ", where $\displaystyle{ \left(X, Y, Z\right) }$ is a 3-tuple of persons; for example, “Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp” is true, while “Karl Marx was introduced to Friedrich Engels by Queen Victoria” is false.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Presentation_of_a_group Retrieved:2015-4-28.
• In mathematics, one method of defining a group is by a 'presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators. We then say G has presentation : $\displaystyle{ \langle S \mid R\rangle. }$ Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

As a simple example, the cyclic group of order n has the presentation : $\displaystyle{ \langle a \mid a^n = 1\rangle. }$ where 1 is the group identity. This may be written equivalently as : $\displaystyle{ \langle a \mid a^n\rangle, }$ since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that include an equals sign.

Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.

A closely related but different concept is that of an absolute presentation of a group.

2009

• WordNet.
• an abstraction belonging to or characteristic of two entities or parts together
• http://en.wiktionary.org/wiki/relation
• The manner in which two things may be associated; A member of one's family; The act of relating a story; A set of ordered tuples; equivalently, a ...

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Arity
• In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product. The term springs from such words as unary, binary, ternary, etc.

The term "arity" is primarily used with reference to operations. If f is the function f : Sn → S, where S is some set, then f is an operation and n is its arity.

• http://sigma.ontologyportal.org:4010/sigma/Browse.jsp?lang=EnglishLanguage&kb=SUMO&term=Relation
• "The Class of relations. There are three kinds of Relation: Predicate, Function, and List. Predicates and Functions both denote sets of ordered n-tuples. The difference between these two Classes is that Predicates cover formula-forming operators, while Functions cover term-forming operators. A List, on the other hand, is a particular ordered n-tuple."
• CYC Glossary http://www.cyc.com/cycdoc/ref/glossary.html
• relation: In Cyc® "relation" is informally used to refer to predicates and functions. In the math or database worlds, a relation is a set of ordered n-tuples. One might talk about the relation "Father", whose elements include (Katherine, Lloyd), (Karen, Wes), (John, Bob), and so on, where the first item in each element is a person and the second is that person's biological father. CycL relations are also ordered n-tuples.

1984

• (Sowa, 1984) ⇒ J. F. Sowa. (1984). “Conceptual Structures: Information Processing in Mind and Machine.
• QUOTE: A relation is a function of one or more arguments whose range is the set of 'truth values' {true,false}. An example of a dyadic or binary relation is the function less than represented by the symbol '<'. Its domain is the set of integers.