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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 A^{2} = A × A. 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.
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.
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 ⊆ A_{1} × … × A_{n}, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A_{j} 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).
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.
A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and G is called its graph.
The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation corresponds to viewing R as the characteristic function on X × Y for the set of pairs of G.
The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independently of each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.
A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.^{[1]} In this case the relation from X to Y is the subset G of X×Y, and "from X to Y" must always be either specified or implied by the context when referring to the relation. In practice correspondence and relation tend to be used interchangeably.
According to the definition above, two relations with identical graphs but different domains or different codomains are considered different. For example, if G = \{(1,2),(1,3),(2,7)\}, then (\mathbb{Z},\mathbb{Z}, G), (\mathbb{R}, \mathbb{N}, G), and (\mathbb{N}, \mathbb{R}, G) are three distinct relations, where \mathbb{Z} is the set of integers and \mathbb{R} is the set of real numbers.
Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations with their graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least one y such that (x,y) \in R, the range of R is defined as the set of all y such that there exists at least one x such that (x,y) \in R, and the field of R is the union of its domain and its range.^{[2]}^{[3]}^{[4]}
A special case of this difference in points of view applies to the notion of function. Many authors insist on distinguishing between a function's codomain and its range. Thus, a single "rule," like mapping every real number x to x^{2}, can lead to distinct functions f: \mathbb{R} \rightarrow \mathbb{R} and f: \mathbb{R} \rightarrow \mathbb{R}^+, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets of ordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As an example, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees it as a relationship that functions may bear to sets.
Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation, and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the two definitions usually matters only in very formal contexts, like category theory.
Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing. Then the binary relation "is owned by" is given as
Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of ordered pairs of the form (object, owner).
The pair (ball, John), denoted by _{ball}R_{John} means that the ball is owned by John.
Two different relations could have the same graph. For example: the relation
is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.
Nevertheless, R is usually identified or even defined as G(R) and "an ordered pair (x, y) ∈ G(R)" is usually denoted as "(x, y) ∈ R".
Some important types of binary relations R between X and Y are listed below.
Uniqueness properties:
Totality properties:
Uniqueness and totality properties:
If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X. Some types of endorelations are widely studied in graph theory, where they are known as simple directed graphs permitting loops.
The set of all binary relations Rel(X) on a set X is the set 2^{X × X} which is a Boolean algebra augmented with the involution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.
Some important properties of a binary relation R over a set X are:
A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is called a total order, simple order, linear order, or a chain.^{[14]} A linear order where every nonempty subset has a least element is called a well-order. A relation that is symmetric, transitive, and serial is also reflexive.
If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y:
If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relation over X and Z: (see main article composition of relations)
A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R y always implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is contained in ≥.
If R is a binary relation over X and Y, then the following is a binary relation over Y and X:
If R is a binary relation over X, then each of the following is a binary relation over X:
If R is a binary relation over X and Y, then the following too:
The complement of the inverse is the inverse of the complement.
If X = Y the complement has the following properties:
The complement of the inverse has these same properties.
The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x and y are in S.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.
Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to the set of rational numbers.
The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain (codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.
Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction =_{A} instead of =. Similarly, the "subset of" relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted ⊆_{A}. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ∈_{A} that is a set. Bertrand Russell has shown that assuming ∈ to be defined on all sets leads to a contradiction in naive set theory.
Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the function with its graph in this context.)^{[15]} With this definition one can for instance define a function relation between every set and its power set.
Notes:
The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).
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