In mathematics, a binary relation over two sets A and B is a set of ordered pairs (a, b) consisting of elements a of A and elements b of B. That is, it is a subset of the Cartesian product . It encodes the information of relation: an element a is related to an element b if and only if the pair (a, b) belongs to the set.
An example is the "divides" relation over the set of prime numbers P and the set of integers 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 2 is related to numbers such as -4, 0, 6, 10, but not 1 or 9, and the prime 3 is related to 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. A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.
A binary relation is the special case of an n-ary relation , 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 of a ternary relation over Z is "... lies between ... and ...", which contains triples such as , , and .
A binary relation over A and B is an element of the power set of . Since the latter set is ordered by inclusion (?), each relation has a place in the lattice of subsets of .
As part of set theory, relations are manipulated with the algebra of sets, including complementation. Furthermore, the two sets are considered symmetrically by introduction of the converse relation which exchanges their places. Another operation is composition of relations. Altogether these tools form the 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,^{[1]}dyadic relation and two-place relation are synonyms for binary relation. But some authors use the term "binary relation" for any subset of a Cartesian product without reference to A and B while the term "correspondence" is reserved for a binary relation with reference to A and B.
Given two sets X and Y, the Cartesian product X × Y is defined as , and its elements are called ordered pairs.
A binary relation R on X and Y is a subset of ; that is, it is a set of ordered pairs consisting of elements and .^{[2]}^{[note 1]} The set X is called the set of departure and the set Y the set of destination or codomain. (In order to specify the choices of the sets X and Y, some authors define a binary relation or a correspondence as an ordered triple where R is a subset of .) The statement is read "x is R-related to y", and is denoted by xRy.
When , a binary relation is called a homogeneous relation. To emphasize the fact X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.^{[3]}^{[4]}^{[5]} An example of a homogeneous relation is a kinship where the relations are over people. Homogeneous relation may be viewed as directed graphs, and in the symmetric case as ordinary graphs. Homogeneous relations also encompass orderings as well as partitions of a set (called equivalence relations).
The order of the elements is important; if then aRb and bRa can be true or false independently of each other. For example, 3 divides 9, but 9 does not divide 3.
The domain of R is the set of all x such that xRy for at least one y. The range or image of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain and its range.^{[6]}^{[7]}^{[8]}
Some authors also call a binary relation a multivalued function;^{[]} in fact, a (single-valued) partial function from X to Y is nothing but a binary relation over X and Y such that for all x in X and y, z in Y.
ball | car | doll | cup | |
---|---|---|---|---|
John | + | - | - | - |
Mary | - | - | + | - |
Venus | - | + | - | - |
ball | car | doll | cup | |
---|---|---|---|---|
John | + | - | - | - |
Mary | - | - | + | - |
Ian | - | - | - | - |
Venus | - | + | - | - |
The following example shows that the choice of codomain is important. Suppose there are four objects and four people . A possible relation on A and B is "is owned by", given by . That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of , i.e. a relation over A and {John, Mary, Venus}.
Some important types of binary relations R over two sets X and Y are listed below.
Uniqueness properties:
Totality properties (only definable if the sets of departure X resp. destination Y are specified):
Uniqueness and totality properties:
If R and S are binary relations over two sets 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 over two sets X and Y is said to be contained in a relation S over X and Y if R is a subset of S, that is, for all x in X and y in Y, if xRy then xSy. 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 in then it has a complementary relation S defined as .
An overline or bar is used to indicate the complementary relation: Alternatively, a strikethrough is used to denote complements, for example, = and ? are complementary to each other, as are ? and ?, and ? and ?. Some authors even use and .^{[]} In total orderings < and >= are complements, as are > and
The complement of the converse relation R^{T} is the converse of the complement:
If , the complement has the following properties:
The restriction of a binary relation over a set X to a subset S is the set of all pairs 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, over the real numbers a property of the relation non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation
The left-restriction (right-restriction, respectively) of a binary relation over two sets X and Y to a subset S of its domain (codomain) is the set of all pairs in the relation for which x (y) is an element of S.
Binary relations over two sets X and Y can be represented algebraically by matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND), matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X and Y and a relation over Y and Z),^{[13]} the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. If X equals Y then the endorelations form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring), and the identity matrix corresponds to the identity relation.^{[14]}
Certain mathematical "relations", such as "equal to", "subset of", and "member 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 over 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 , as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.)^{[15]} With this definition one can for instance define a binary relation over every set and its power set.
A homogeneous relation (also called endorelation) over a set X is a binary relation over the set X and itself, i.e. it is a subset of the Cartesian product .^{[5]}^{[16]}^{[17]} It is also simply called a binary relation over X.
A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, where X is the vertex set and R is the edge set (there is an edge from a vertex x to a vertex y if and only if xRy). The homogenous relation is called the adjacency relation of the directed graph.
The set of all binary relations over 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 converse relation. Considering composition of relations as a binary operation on it forms a inverse semigroup.
Some important particular binary relations over a set X are:
For arbitrary elements x and y of X,
Some important properties that a binary relation R over a set X may have are:
The previous 4 alternatives are far from being exhaustive; e.g., the red relation from the above picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair , and , but not , respectively. The latter two facts also rule out quasi-reflexivity.
Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by is neither symmetric nor antisymmetric, let alone asymmetric.
A preorder is a relation that is reflexive and transitive. A total preorder, also called weak order, is a relation that is reflexive, transitive, and connex. A partial order is a relation that is reflexive, antisymmetric, and transitive. A total order, also called linear order, simple order, connex order, or chain is a relation that is reflexive, antisymmetric, transitive and connex.^{[26]}
A partial equivalence relation is a relation that is symmetric and transitive. An equivalence relation is a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.
If R is a homogeneous relation over X then each of the following is a homogeneous relation over X:
All operations defined in the above section #Operations on binary relations also apply to homogeneous relations.
Reflexivity | Symmetry | Transitivity | Symbol | Example | |
---|---|---|---|---|---|
Directed graph | -> | ||||
Undirected graph | Symmetric | ||||
Tournament | Irreflexive | Antisymmetric | Pecking order | ||
Dependency | Reflexive | Symmetric | |||
Preorder | Reflexive | Yes | Preference | ||
Strict preorder | Irreflexive | Yes | < | ||
Total preorder | Reflexive | Yes | |||
Partial order | Reflexive | Antisymmetric | Yes | Subset | |
Strict partial order | Irreflexive | Antisymmetric | Yes | < | Proper subset |
Strict weak order | Irreflexive | Antisymmetric | Yes | < | |
Total order | Reflexive | Antisymmetric | Yes | ||
Partial equivalence relation | Symmetric | Yes | |||
Equivalence relation | Reflexive | Symmetric | Yes | ~, ?, ?, ? | Equality |
The number of distinct binary relations over an n-element set is 2^{n2} (sequence in the OEIS):
Elements | Any | Transitive | Reflexive | Preorder | Partial order | Total preorder | Total order | Equivalence relation |
---|---|---|---|---|---|---|---|---|
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |
3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |
4 | 355 | 219 | 75 | 24 | 15 | |||
n | 2^{n2} | 2^{n2-n} | ?n k=0 k! S(n, k) |
n! | ?n k=0 S(n, k) | |||
OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
Notes:
The binary relations can be grouped into pairs (relation, complement), except that for the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).