Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

In other words, it is a subset of the Cartesian product A2 = A × A.

Generalization

More generally, a binary relation between two sets A and B is a subset of A × B.

The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.

Example

An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p).

In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.

Applications

Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more.

The concept of function is defined as a special kind of binary relation.

Binary relations are also heavily used in computer science.

Special case

A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain Aj of the relation.

An example for a ternary relation on Z×Z×Z is "lies between ... and ...", containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).

See Tuple.

Axiomatic set theory

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets.

This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

See also

• Axiomatic set theory
• Binary
• Computer science
• Confluence (term rewriting)
• Hasse diagram
• Incidence structure
• Logic of relatives
• Mathematics
• Order theory.
• Set (mathematics)
• Triadic relation
• Tuple

External links

• Binary relation @ Wikipedia