Relation (mathematics)

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  • 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 introduction[edit]

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:


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.



Example 1. Divisibility[edit]

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 definitions[edit]

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:

  • 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.


Example 2. Coplanarity[edit]


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.




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.






Focal nodes[edit]

Peer nodes[edit]

Logical operators[edit]

Related topics[edit]

Relational concepts[edit]

Information, Inquiry[edit]

Related articles[edit]

Document history[edit]
