| Category: SetTheory |
mathworld term : BinaryRelation
from WikiPedia:Binary_relation
In mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of another set.
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 no other. 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.
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, and many more. The all-important concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science, especially within the relational model for databases.
A binary relation is a special case of a k-ary relation, that is, a set of k-tuples where the jth component of each k-tuple is taken from the jth domain Xj of the relation. A k-ary relation among elements of a single set is said to be homogeneous.
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.