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Relation (mathematics)

This page belongs to resource collections on Logic and Inquiry.

In mathematics, a finitary relation is defined by one of the formal definitions given below.

  • The basic idea is to generalize the concept of a two-place relation, such as the relation of equality denoted by the sign “” in a statement like or the relation of order denoted by the sign “” in a statement like   Relations that involve two places or roles are called binary relations by some and dyadic relations by others, the latter being historically prior but also useful when necessary to avoid confusion with binary (base 2) numerals.
  • The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.  These are called finite-place or finitary relations.  A finitary relation involving places is variously called a -ary, -adic, or -dimensional relation.  The number is then called the arity, the adicity, or the dimension of the relation, respectively.

Informal introductionEdit

The definition of relation given in the next section formally captures a concept that is actually quite familiar from everyday life.  For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form    The facts of a concrete situation could be organized in the form of a Table like the one below:


 
     
     
     
     
     


Each row of the Table records a fact or makes an assertion of the form    For instance, the first row says, in effect,    The Table represents a relation   over the set   of people under discussion:

 

The data of the Table are equivalent to the following set of ordered triples:

 

By a slight overuse of notation, it is usual to write   to say the same thing as the first row of the Table.  The relation   is a triadic or ternary relation, since there are three items involved in each row.  The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package.

The Table for relation   is an extremely simple example of a relational database.  The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.  Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation.

For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth.

Example 1. DivisibilityEdit

A more typical example of a two-place relation in mathematics is the relation of divisibility between two positive integers   and   that is expressed in statements like   or    This is a relation that comes up so often that a special symbol   is reserved to express it, allowing one to write   for  

To express the binary relation of divisibility in terms of sets, we have the set   of positive integers,   and we have the binary relation   on   such that the ordered pair   is in the relation   just in case    In other turns of phrase that are frequently used, one says that the number   is related by   to the number   just in case   is a factor of   that is, just in case   divides   with no remainder.  The relation   regarded as a set of ordered pairs, consists of all pairs of numbers   such that   divides  

For example,   is a factor of   and   is a factor of   which two facts can be written either as   and   or as   and  

Formal definitionsEdit

There are two definitions of  -place relations that are commonly encountered in mathematics.  In order of simplicity, the first of these definitions is as follows:

Definition 1.   A relation   over the sets   is a subset of their cartesian product, written    Under this definition, then, a  -ary relation is simply a set of  -tuples.

The second definition makes use of an idiom that is common in mathematics, saying that “such and such is an  -tuple” to mean that the mathematical object being defined is determined by the specification of   component mathematical objects.  In the case of a relation   over   sets, there are   things to specify, namely, the   sets plus a subset of their cartesian product.  In the idiom, this is expressed by saying that   is a  -tuple.

Definition 2.   A relation   over the sets   is a  -tuple   where   is a subset of the cartesian product   called the graph of  

Elements of a relation are sometimes denoted by using boldface characters, for example, the constant element   or the variable element  

A statement of the form “  is in the relation  ” is taken to mean that   is in   under the first definition and that   is in   under the second definition.

The following considerations apply under either definition:

  • The sets   for   are called the domains of the relation.  In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains.
  • If all the domains   are the same set   then   is more simply referred to as a  -ary relation over  
  • If any domain   is empty then the cartesian product is empty and the only relation over such a sequence of domains is the empty relation    Most applications of the relation concept will set aside this trivial case and assume that all domains are nonempty.

If   is a relation over the domains   it is conventional to consider a sequence of terms called variables,   that are said to range over the respective domains.

A boolean domain   is a generic 2-element set, say,   whose elements are interpreted as logical values, typically   and  

The characteristic function of the relation   written   or   is the boolean-valued function   defined in such a way that   just in case the  -tuple   is in the relation    The characteristic function of a relation may also be called its indicator function, especially in probabilistic and statistical contexts.

It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like   as a  -place predicate.  From the more abstract viewpoints of formal logic and model theory, the relation   is seen as constituting a logical model or a relational structure that serves as one of many possible interpretations of a corresponding  -place predicate symbol, as that term is used in predicate calculus.

Due to the convergence of many traditions of study, there are wide variations in the language used to describe relations.  The extensional approach presented in this article treats a relation as the set-theoretic extension of a relational concept or term.  An alternative, intensional approach reserves the term relation to the corresponding logical entity, either the logical comprehension, which is the totality of intensions or abstract properties that all the elements of the extensional relation have in common, or else the symbols that are taken to denote those elements and intensions.

Example 2. CoplanarityEdit

For lines   in three-dimensional space, there is a triadic relation picking out the triples of lines that are coplanar.  This does not reduce to the dyadic relation of coplanarity between pairs of lines.

In other words, writing   when the lines   lie in a plane, and   for the binary relation, it is not true that     and   together imply   although the converse is certainly true:  any two of three coplanar lines are necessarily coplanar.  There are two geometrical reasons for this.

In one case, for example taking the  -axis,  -axis, and  -axis, the three lines are concurrent, that is, they intersect at a single point.  In another case,   can be three edges of an infinite triangular prism.

What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

RemarksEdit

Relations are classified by the number of sets in the cartesian product, in other words, the number of places or terms in the relational expression:

  Monadic or unary relation, in other words, a property or set
  Dyadic or binary relation
  Triadic or ternary relation
  Tetradic or quaternary relation
  Pentadic or quinary relation

Relations with more than five terms are usually referred to as  -adic or  -ary, for example, a 6-adic, 6-ary, or hexadic relation.

ReferencesEdit

  • 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, 317–378, 1870. Reprinted, Collected Papers CP 3.45–149, Chronological Edition CE 2, 359–429.
  • 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.

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