# Relation Function

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

**AKA:***R*, Binary-Output Function, Two Value-Output Function.**Context:**- It can range from being a Unary Relation to being a Binary Relation to being an n-Ary Relation, depending on its Arity.
- It can range from being a Typed Relation (associated to a relation type) to being an Untyped Relation.
- It can range from being a Reflexive Relation to being an Irreflexive Relation.
- It can range from being a Symmetric Relation to being an Antisymmetric Relation.
- It can range from being a Transitive Relation to being an Intransitive Relation.
- It can range from being a One-to-One Relation, One-to-Many Relation, or a Many-to-Many Relation.
- It can range from being a Total Relation or a Non-Total Relation (to some given set).
- It can range from being a Surjective Relation to being an Injective Relation.
- It can be associated to a Relation System.
- It can be in an Inverse Relation with another Binary Relation (with switched parameters).
- It can be (instantiated as) a Ground Fact. (InstanceOf Relation?)
- It can be a Semantic Relation (Logic Relation)
- if its Output Set is the Truth Set.
- when the relation is between Concepts.

**Example(s):***z=PositiveNumber(x) ⇒ z=1 if x > 0; else z=0*. (A Unary Relation)*z=GreaterThan(x,y) ⇒ z=TRUE if x>y; else z=FALSE.*(A binary Semantic Relation)*z=Person(x)*⇒*z=TRUE*if [math]\displaystyle{ x }[/math] belongs to the person Entity Type; else*z=FALSE*.*z=OPL(O,P,L)*⇒*z=TRUE*if $O$ refers to an organism, [math]\displaystyle{ P }[/math] refers to a protein, [math]\displaystyle{ L }[/math] refers to a Subcellular Location and [math]\displaystyle{ P }[/math] is a protein of $O$ and localizes in*L*; else*z=FALSE*.- a Category Relation, Numeric Relation, ...
- a Set Relation, Multiset Relation, ...
- a Sequence Relation, String Relation, ...
- a Tuple Relation, Vector Relation, ...
- a Graph Relation, Tree Relation, ...
- …

**Counter-Example(s):**- a Continuous Function.
- a Binary Function.
- an Entity, such as a Data Value.
- a Binary-Input Function (whose Function Output is not a Binary Set).

**See:**Relationship, Set Operation, Formula Variable, Relational Calculus, Relational Table, Relator, Russell's Paradox, Ordered Pair, Cartesian Product.

## 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 [math]\displaystyle{ k }[/math] -tuples of individuals. Typically, the property describes a possible connection between the components of a [math]\displaystyle{ k }[/math] -tuple. For a given set of [math]\displaystyle{ k }[/math] -tuples, a truth value is assigned to each [math]\displaystyle{ k }[/math] -tuple according to whether the property does or does not hold.An example of a

*ternary relation*(i.e., between three individuals) is: " [math]\displaystyle{ X }[/math] was introduced to [math]\displaystyle{ Y }[/math] by [math]\displaystyle{ Z }[/math] ", where [math]\displaystyle{ \left(X, Y, Z\right) }[/math] 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.

- In mathematics, a

### 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*R**generators**so that every element of the group can be written as a product of powers of some of these generators, and a set*of***relations**among those generators. We then say**G***has presentation : [math]\displaystyle{ \langle S \mid R\rangle. }[/math] 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*.*relators, distinguishing them from the relations that include an equals sign.*As a simple example, the cyclic group of order*n has the presentation : [math]\displaystyle{ \langle a \mid a^n = 1\rangle. }[/math] where 1 is the group identity. This may be written equivalently as : [math]\displaystyle{ \langle a \mid a^n\rangle, }[/math] since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are calledEvery 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.

- In mathematics, one method of defining a group is by a '

### 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.

- 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.
- 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.

### 1977

- (Makinouchi, 1977) ⇒ Akifumi Makinouchi. (1977). “A Consideration on Normal Form of Not-Necessarily-Normalized Relation in the Relational Data Model.” In: Proceedings of the third International Conference on Very large data bases (VLDB 1977).
- QUOTE: Mathematically, a relation is defined as follows: Given sets [math]\displaystyle{ D_1, D_2, ..., D_n }[/math], (not necessarily distinct), [math]\displaystyle{ R }[/math] is a relation on these [math]\displaystyle{ n }[/math] sets if it is a set of n-tuples ([math]\displaystyle{ d_1, d_2, ..., d_n }[/math]) such that [math]\displaystyle{ d_i }[/math] belongs to [math]\displaystyle{ D_i }[/math], where [math]\displaystyle{ i=l, 2, ..., n }[/math]. [math]\displaystyle{ D_1, D_2, ..., D_n }[/math] are domains of [math]\displaystyle{ R }[/math]. In the realm of relational data model, a relation as above looks like a table (array) each of whose columns has different names. In the sequel, we use tables and relations interchangeably, but domains and columns differently. Each column name of a relation represents a role name in the relation and its domain is a set of values which may possibly be inserted into the column. Each row of the relation consists of an n-tuple of values.

### 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.

- QUOTE: A relation is a function of one or more arguments whose range is the set of 'truth values' {