Binary relation

related topics
{math, number, function}
{car, race, vehicle}
{woman, child, man}
{group, member, jewish}
{ship, engine, design}
{style, bgcolor, rowspan}

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. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.

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 (and not with any 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.

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.

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.

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.

Contents

Full article ▸

related documents
Pushdown automaton
Abstract interpretation
E (mathematical constant)
Axiom schema of replacement
Heine–Borel theorem
Random variable
Euclidean algorithm
Glossary of topology
Algebraic structure
Linear combination
Presentation of a group
Factorization
Linear independence
Ideal class group
Type theory
Sequence
Elliptic curve
IEEE 754-1985
Gaussian quadrature
Natural transformation
Fuzzy logic
Absolute convergence
Galois theory
Probability density function
Partition (number theory)
Dijkstra's algorithm
Ultrafilter
Database normalization
Euler characteristic
Complex analysis