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In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a mathematical function, but with fewer restrictions. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1,..., Xn, which is a subset of the Cartesian product X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.}
An example of a binary relation is the "divides" relation over the set of prime numbers P {\displaystyle \mathbb {P} } and the set of integers Z {\displaystyle \mathbb {Z} } , in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.
Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.