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, Characteristic Function.
• Context:
  • It can typically map input tuple to truth value through formal evaluation.
  • It can typically represent logical property through binary output.
  • It can typically establish validity assertion for relationship between entitys.
  • It can typically formalize membership test for mathematical sets.
  • It can typically enable formal verification through truth condition evaluation.
  • ...
  • It can often support logical reasoning through inference rule application.
  • It can often serve as foundation for knowledge representation systems.
  • It can often express constraint on domain entitys.
  • It can often facilitate query processing in database systems.
  • It can often model entity relationship in information systems.
  • ...
  • It can range from being a Unary Relation Function to being a Binary Relation Function to being an n-Ary Relation Function, depending on its relation function arity.
  • It can range from being a Typed Relation Function to being an Untyped Relation Function, depending on its relation function type specification.
  • It can range from being a Reflexive Relation Function to being an Irreflexive Relation Function, depending on its relation function self-reference property.
  • It can range from being a Symmetric Relation Function to being an Antisymmetric Relation Function, depending on its relation function directionality property.
  • It can range from being a Transitive Relation Function to being an Intransitive Relation Function, depending on its relation function chain property.
  • It can range from being a One-to-One Relation Function to being a Many-to-Many Relation Function, depending on its relation function cardinality property.
  • It can range from being a Total Relation Function to being a Non-Total Relation Function, depending on its relation function completeness property.
  • It can range from being a Surjective Relation Function to being an Injective Relation Function, depending on its relation function mapping property.
  • It can range from being a Domain-Independent Relation Function to being a Domain-Specific Relation Function, depending on its relation function application scope.
  • It can range from being a Concrete Relation Function to being an Abstract Relation Function, depending on its relation function abstraction level.
  • ...
  • It can be associated to a Relation System for systematic relation processing.
  • It can be in an Inverse Relation with another Relation Function through parameter order reversal.
  • It can be instantiated as a Ground Fact in knowledge bases.
  • It can be formalized in Predicate Logic as a predicate expression.
  • It can be implemented in Relational Database as a relational constraint.
  • It can be defined over Continuous Domains or Discrete Domains.
  • It can be represented as a Set of Tuples from a cartesian product.
  • It can be visualized as a Directed Graph where edges represent relationships.
  • It can be a Semantic Relation when its Output Set is the Truth Set and the relation is between Concepts.
  • It can be expressed using Formal Notation, Mathematical Symbols, or Logical Expressions.
  • It can be stored in a Relational Table where each row represents a valid relation instance.
  • It can be evaluated for properties such as Symmetry, Transitivity, and Reflexivity.
  • ...
• Examples:
  • Relation Function Mathematical Implementations, such as:
    • Logical Relation Functions, such as:
      • Equality Relation Function for value sameness test.
      • Inequality Relation Function for value difference test.
      • Subset Relation Function for set containment test.
      • Membership Relation Function for element inclusion test.
    • Numeric Relation Functions, such as:
      • GreaterThan Relation Function for magnitude comparison.
      • LessThan Relation Function for magnitude comparison.
      • Divisibility Relation Function for integer factor test.
      • Positive Number Relation Function for sign test.
    • Structural Relation Functions, such as:
      • Adjacency Relation Function for node connection test.
      • Path Relation Function for reachability test.
      • Ancestor Relation Function for hierarchical precedence test.
      • Containment Relation Function for spatial inclusion test.
  • Relation Function Application Domains, such as:
    • Set-Based Relation Functions, such as:
      • Set Relation Function for set element relationships.
      • Multiset Relation Function for multiset element relationships.
      • Category Relation Function for category membership.
      • Collection Relation Function for collection membership.
    • Sequence-Based Relation Functions, such as:
      • Sequence Relation Function for sequential element relationships.
      • String Relation Function for character pattern relationships.
      • Time Series Relation Function for temporal pattern relationships.
      • Ordering Relation Function for element position relationships.
    • Structure-Based Relation Functions, such as:
      • Tuple Relation Function for component relationships.
      • Vector Relation Function for dimensional relationships.
      • Graph Relation Function for network relationships.
      • Tree Relation Function for hierarchical relationships.
  • Relation Function Practical Applications, such as:
    • Database Relation Functions, such as:
      • Foreign Key Relation Function for referential integrity.
      • Constraint Relation Function for data validity checking.
      • Query Relation Function for record selection.
      • Join Relation Function for table combination.
    • Domain-Specific Relation Functions, such as:
      • Person Relation Function for human entity classification.
      • Semantic Web Relation Function for resource relationships.
      • OPL Relation Function for biological localization relationships.
      • Spatial Relation Function for geographic relationships.
  • ...
• Counter-Examples:
  • Continuous Function, which maps to continuous range rather than binary set.
  • Binary Function, which takes two inputs but produces non-binary output.
  • Entity, such as a Data Value, which is not a function but an object.
  • Binary-Input Function, whose function output is not a binary set.
  • Mathematical Operation, which combines values rather than testing relationships.
  • Data Structure, which organizes information rather than evaluating relationships.
  • Scalar Function, which returns a value from a non-binary range.
  • Vector Function, which returns a multi-component result.
• See: Relationship, Set Operation, Formula Variable, Relational Calculus, Relational Table, Relator, Russell's Paradox, Ordered Pair, Cartesian Product, Predicate Logic, First Order Logic, Truth Function, Characteristic Function, Boolean Function, Knowledge Representation, Semantic Network, Formal System, Database Theory, Relational Algebra, Set Theory, Graph Theory.

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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.

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 : [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.

    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 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.

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.

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