dbo:abstract

 In mathematics, a binary relation is a general concept that defines some relation between the elements of two sets. It is a generalization of the more commonly understood idea of a mathematical function, but with fewer restrictions. A binary relation over sets X and Y is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) 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 nary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X1 × ... × Xn. A trivial example of a binary relation over set X of all real numbers and set Y of all real numbers is the set of all pairs for which elements . This is equivalent to the function . Another example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers , 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:
* the "is greater than", "is equal to", and "divides" relations in arithmetic;
* the "is congruent to" relation in geometry;
* the "is adjacent to" relation in graph theory;
* the "is orthogonal to" relation in linear algebra. A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science, such as in a relational database management system (RDBMS). A binary relation over sets X and Y is an element of the power set of X × Y. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X × Y. A binary relation is either a or a heterogeneous relation depending on whether X = Y or not. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice. 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. The terms correspondence, dyadic relation and twoplace relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y. (en)

rdfs:comment

 In mathematics, a binary relation is a general concept that defines some relation between the elements of two sets. It is a generalization of the more commonly understood idea of a mathematical function, but with fewer restrictions. A binary relation over sets X and Y is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) 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 nary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X1 × ... × Xn. (en)
