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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a finitary relation over sets is a subset of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
; that is, it is a set of ''n''-tuples consisting of elements ''x''''i'' in ''X''''i''. 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'' giving the number of "places" in the relation is called the ''
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...
'', ''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 infinite sequences. An ''n''-ary relation over sets is an element of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of . 0-ary relations count only two members: the one that always holds, and the one that never holds. This is because there is only one 0-tuple, the empty tuple (). They are sometimes useful for constructing the base case of an induction argument. Unary relations can be viewed as a collection of members (such as the collection of
Nobel laureates The Nobel Prizes ( sv, Nobelpriset, no, Nobelprisen) are awarded annually by the Royal Swedish Academy of Sciences, the Swedish Academy, the Karolinska Institutet, and the Norwegian Nobel Committee to individuals and organizations who make ou ...
) having some property (such as that of having been awarded the
Nobel prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
).
Binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s are the most commonly studied form of finitary relations. When ''X''1 = ''X''2 it is called a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
, for example: * Equality and inequality, denoted by signs such as = and < in statements such as "", or *
Divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
, denoted by the sign , in statements such as "13, 143". Otherwise it is a
heterogeneous relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
, for example: *
Set membership In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets o ...
, denoted by the sign ∈ in statements such as "".


Example

Consider the ternary relation ''R'' "''x'' thinks that ''y'' likes ''z''" over the set of people , defined by: : . ''R'' can be represented equivalently by the following table: Here, each row represents a triple of ''R'', that is it makes a statement of the form "''x'' thinks that ''y'' likes ''z''". For instance, the first row states that "Alice thinks that Bob likes Denise". All rows are distinct. The ordering of rows is insignificant but the ordering of columns is significant. The above table is also a simple example of a
relational database A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relati ...
, a field with theory rooted in relational algebra and applications in data management. Computer scientists, logicians, and mathematicians, however, tend to have different conceptions what a general relation is, and what it is consisted of. For example, databases are designed to deal with empirical data, which is by definition finite, whereas in mathematics, relations with infinite arity (i.e., infinitary relation) are also considered.


Definitions

The first definition of relations encountered in mathematics is: ; Definition 1: An ''n''-ary relation ''R'' over sets is a subset of the Cartesian product . The second definition of relations makes use of an idiom that is common in mathematics, stipulating that "such and such is an ''n''-tuple" in order to ensure that such and such a mathematical object is determined by the specification of mathematical objects with ''n'' elements. In the case of a relation ''R'' over ''n'' sets, there are things to specify, namely, the ''n'' sets plus a subset of their Cartesian product. In the idiom, this is expressed by saying that ''R'' is a ()-tuple. ; Definition 2: An ''n''-ary relation ''R'' over sets is an ()-tuple where ''G'' is a subset of the Cartesian product called the ''graph'' of ''R''. As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and if it ever becomes necessary to distinguish between the two definitions, then an entity satisfying the second definition may be called an ''embedded'' or ''included relation''. Both statements (under the first definition) and (under the second definition) read "''x''1, ⋯, ''x''''n'' are ''R''-related" and are denoted using prefix notation by and using
postfix notation Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in w ...
by . In the case where ''R'' is a binary relation, those statements are also denoted using
infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in . Usage Binary relations are ...
by . The following considerations apply under either definition: * The set ''X''''i'' is called the th ''domain'' of ''R''. Under the first definition, the relation does not uniquely determine a given sequence of domains. In the case where ''R'' is a binary relation, ''X''1 is also called simply the ''domain'' or ''set of departure'' of ''R'', and ''X''2 is also called the ''codomain'' or ''set of destination'' of ''R''. * When the elements of ''X''''i'' are relations, ''X''''i'' is called a ''nonsimple domain'' of ''R''. * The set of for which there exists such that is called the ''i''th ''domain of definition'' or ''active domain'' of ''R''. In the case where ''R'' is a binary relation, its first domain of definition is also called simply the ''domain of definition'' or ''active domain'' of ''R'', and its second domain of definition is also called the ''codomain of definition'' or ''active codomain'' of ''R''. * When the th domain of definition of ''R'' is equal to ''X''''i'', ''R'' is said to be ''total'' on ''X''''i''. In the case where ''R'' is a binary relation, when ''R'' is total on ''X''1, it is also said to be ''left-total'' or ''serial'', and when ''R'' is total on ''X''2, it is also said to be ''right-total'' or ''surjective''. * When , where , , , and is a partition of , ''R'' is said to be ''unique'' on , and is called ''a
primary key In the relational model of databases, a primary key is a ''specific choice'' of a ''minimal'' set of attributes ( columns) that uniquely specify a tuple ( row) in a relation ( table). Informally, a primary key is "which attributes identify a recor ...
'' of ''R''. In the case where ''R'' is a binary relation, when ''R'' is unique on , it is also said to be ''left-unique'' or ''injective'', and when ''R'' is unique on , it is also said to be ''right-unique'' or ''functional''. * When all ''X''''i'' are the same set ''X'', it is simpler to refer to ''R'' as an ''n''-ary relation over ''X'', called a ''homogeneous relation''. Otherwise ''R'' is called a ''heterogeneous relation''. * When any of ''X''''i'' is empty, the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation . Hence it is commonly stipulated that all of the domains be nonempty. Let a
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as ...
B be a two-element set, say, , whose elements can be interpreted as logical values, typically and . The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of ''R'', denoted by χ''R'', is the Boolean-valued function , defined by if and otherwise. In applied mathematics,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and statistics, it is common to refer to a Boolean-valued function as an ''n''-ary ''predicate''. From the more abstract viewpoint of
formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
and
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, the relation ''R'' constitutes a ''logical model'' or a ''relational structure'', that serves as one of many possible
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...
s of some ''n''-ary predicate symbol. Because relations arise in many scientific disciplines, as well as in many branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, there is considerable variation in terminology. Aside from the
set-theoretic Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...
of a relational concept or term, the term "relation" can also be used to refer to the corresponding logical entity, either the logical comprehension, which is the totality of
intension In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or ano ...
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).


History

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990).
Charles Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
,
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
and others advanced the theory of relations. Many of their ideas, especially on relations called
orders Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
, were summarized in '' The Principles of Mathematics'' (1903) where
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
made free use of these results. In 1970,
Edgar Codd Edgar Frank "Ted" Codd (19 August 1923 – 18 April 2003) was an English computer scientist who, while working for IBM, invented the relational model for database management, the theoretical basis for relational databases and relational datab ...
proposed a
relational model The relational model (RM) is an approach to managing data using a structure and language consistent with first-order predicate logic, first described in 1969 by English computer scientist Edgar F. Codd, where all data is represented in terms of t ...
for
database In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases ...
s, thus anticipating the development of
data base management system In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases spa ...
s.


See also

* Incidence structure *
Hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) w ...
*
Logic of relatives Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for t ...
*
Logical matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix (mathematics), matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. ...
*
Partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
*
Predicate (mathematical logic) In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula P(a), the symbol P is a predicate which applies to the individual constant a. Similarly, in the formula R(a,b), R is a predicat ...
* Projection (set theory) *
Reflexive relation In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
*
Relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
* Relational algebra *
Relational model The relational model (RM) is an approach to managing data using a structure and language consistent with first-order predicate logic, first described in 1969 by English computer scientist Edgar F. Codd, where all data is represented in terms of t ...
* Relations (philosophy)


References


Bibliography

* * Bourbaki, N. (1994) ''Elements of the History of Mathematics'', John Meldrum, trans. Springer-Verlag. * Carnap, Rudolf (1958) ''Introduction to Symbolic Logic with Applications''. Dover Publications. * Halmos, P.R. (1960) ''Naive Set Theory''. Princeton NJ: D. Van Nostrand Company. * Lawvere, F.W., and R. Rosebrugh (2003) ''Sets for Mathematics'', Cambridge Univ. Press. * Lewis, C.I. (1918) A Survey of Symbolic Logic, Chapter 3: Applications of the Boole—Schröder Algebra, via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
* Lucas, J. R. (1999) ''Conceptual Roots of Mathematics''. Routledge. * Maddux, R.D. (2006) ''Relation Algebras'', vol. 150 in "Studies in Logic and the Foundations of Mathematics". Elsevier Science. * Merrill, Dan D. (1990) ''Augustus De Morgan and the logic of relations''. Kluwer. * Peirce, C.S. (1870), "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–78, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429. * Peirce, C.S. (1984) ''Writings of Charles S. Peirce: A Chronological Edition, Volume 2, 1867-1871''. Peirce Edition Project, eds. Indiana University Press. * Russell, Bertrand (1903/1938)
The Principles of Mathematics, 2nd ed.
' Cambridge Univ. Press. * Suppes, Patrick (1960/1972) ''Axiomatic Set Theory''. Dover Publications. * Tarski, A. (1956/1983) ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger, trans. 1st edition, Oxford University Press. 2nd edition, J. Corcoran, ed. Indianapolis IN: Hackett Publishing. * Ulam, S.M. and Bednarek, A.R. (1990), "On the Theory of Relational Structures and Schemata for Parallel Computation", pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA. * Ulam, S.M. (1990) ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'' in A.R. Bednarek and Françoise Ulam, eds., University of California Press. *
Roland Fraïssé Roland Fraïssé (; 12 March 1920 – 30 March 2008) was a French mathematical logician. Fraïssé received his doctoral degree from the University of Paris in 1953. In his thesis, Fraïssé used the back-and-forth method to determine whether ...
(2000) 986''Theory of Relations'', North Holland {{Authority control Mathematical logic Mathematical relations