Binary relation

"Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For other uses, see Relation (disambiguation).

This article is about a binary relation between two sets, possibly distinct. For a binary relation on a single set (a special case), see #Homogeneous relation.

In mathematics, a binary relation between two sets A and B is a set of ordered pairs (a, b) consisting of elements a of A and elements b of B; in short, it is a subset of the Cartesian product A × B. 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 between 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 that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is related to 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. 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 n = 2 of an n-ary relation R ⊆ A₁ × … × 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).

A binary relation on A × B is an element in the power set on A × B. Since the latter set is ordered by inclusion (⊂), each relation has a place in the lattice of subsets of A × B. A binary relation between the same set is also called a homogeneous relation (and a binary relation is sometimes called a heterogeneous relation to emphasize the fact it is not necessarily homogeneous). An example of a homogeneous relation is a kinship where the relations are between 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).

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, dyadic relation and 2-place relation are synonyms for binary relation. But some authors use the term “binary relation” for any subset of a Cartesian product A × B without the reference to A, B while the term “correspondence” is reserved for a binary relation with the reference to A, B.

Definition

Given a pair of sets X, Y, there is the set called the Cartesian product $X\times Y=\{(x,y)|x\in X,y\in Y\}$, whose elements are called ordered pairs.

A binary relation R from X to Y is a subset of $X\times Y$; that is, it is a set of ordered pairs $(x,y)$ consisting of elements $x\in X$ and $y\in Y$. The set $X$ is called the set of departure and the set Y, the set of destination or codomain. A binary relation is also called a correspondence. (In order to specify the choices of the sets $X,Y$, some authors define a binary relation or a correspondence as an ordered triple $(X,Y,R)$ where $R$ is a subset of $X\times Y$.)

When $X=Y$, a binary relation is called a homogeneous relation. To emphasize the fact $X,Y$ are allowed to be different, a binary relation is also called a heterogeneous relation.

The statement $(x,y)\in R$ is read "x is R-related to y", and is denoted by xRy.

The order of the elements in each pair of R is important: if a ≠ b, then aRb and bRa can be true or false, independently of each other. Resuming the example in the lead, the prime 3 divides the integer 9, but 9 doesn't divide 3.

The domain of R is the set of all x such that xRy for at least one y. The range 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.

A binary relation is also called a multivalued function; in fact, a (single-valued) function is nothing but a binary relation such that $xRy,xRy'\Rightarrow y=y'$.

Example

2nd example relation

	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Venus	−	+	−	−

1st example relation

	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Ian	−	−	−	−
Venus	−	+	−	−

The following example shows a choice of codomain matters (and thus is a part of a definition of a relation).

Suppose there are four objects A = {ball, car, doll, cup} and four persons B = {John, Mary, Ian, Venus}. A possible example of "is owned by" is:

R = { (ball, John), (doll, Mary), (car, Venus) }.

That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing.

Now, as a set, R involves no Ian; hence, R could have been viewed as a subset of A × {John, Mary, Venus}. But that will encode different information; namely, it does not tell anything about ownership of Ian.

Special types of binary relations

Example relations between real numbers. Red: y=x². Green: y=2x+20.

Some important types of binary relations R between two sets X and Y are listed below.

Uniqueness properties:

• injective (also called left-unique): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. For example, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5 and z = +5 to y = 25.
• functional (also called univalent or right-unique or right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations in the picture are functional. An example for a non-functional relation can be obtained by rotating the red graph clockwise by 90 degrees, i.e. by considering the relation x=y² which relates e.g. x=25 to both y=-5 and z=+5.
• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties (only definable if the sets of departure X resp. destination Y are specified):

• left-total: for all x in X there exists a y in Y such that xRy. For example, R is left-total when it is a function or a multivalued function. Note that this property, although sometimes also referred to as total, is different from the definition of total in the next section. Both relations in the picture are left-total. The relation x=y², obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.
• surjective (also called right-total or onto): for all y in Y there exists an x in X such that xRy. The green relation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.
• An injective function or injection: a relation that is injective, functional, and left-total.
• A surjective function or surjection: a relation that is functional, left-total, and right-total.
• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known as one-to-one correspondence. The green relation is bijective, but the red is not.

Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y:

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. The identity element is the empty relation. For example, ≥ is the union of > and =.
• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }. The identity element is the universal relation.

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)

• Composition: S ∘ R, also denoted R ; S (or R ∘ S), defined as S ∘ R = { (x, z) | there exists y ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The identity element is the identity relation. The order of R and S in the notation S ∘ R, used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" ∘ "is parent of" yields "is maternal grandparent of", while the composition "is parent of" ∘ "is mother of" yields "is grandmother of".

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:

• Converse: R ^T, defined as R ^T = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to its converse if and only if it is symmetric. See also duality (order theory). For example, "is less than" (<) is the converse of "is greater than" (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R ⁼, defined as R ⁼ = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
• Reflexive reduction: R ^≠, defined as R ^≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over X contained in R.
• Transitive closure: R ⁺, defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
• Reflexive transitive closure: R *, defined as R * = (R ⁺) ⁼, the smallest preorder containing R.
• Reflexive transitive symmetric closure: R ^≡, defined as the smallest equivalence relation over X containing R.

Complement

If R is a binary relation in X × Y, then it has a

• complementary relation S is defined as x S y if not x R y. An overline or bar is used to indicate the complementary relation: $S\ =\ {\bar {R}}.$ 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 $R$ and $\not R$.

In total orderings < and ≥ are complements, as are > and ≤.

The complement of the converse relation R^T is the converse of the complement:${\overline {R^{T}}}\ =\ {\bar {R}}^{T}.$

If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.
• The complement of a reflexive relation is irreflexive and vice versa.
• The complement of a strict weak order is a total preorder and vice versa.

Restriction

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.

Matrix representation

Binary relations between 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 between X and Y and a relation between Y and Z), 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.

Sets versus classes

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.) With this definition one can for instance define a function relation between every set and its power set.

Homogeneous relation

A homogeneous relation on a set X is a binary relation between the same set; i.e., it is a subset of a Cartesian product $X\times X$. It is also called a binary relation 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.

A homogeneous relation on a set X may be identified with a directed graph, where X is the set of (possibly infinitely many) vertices and there is an edge from a vertex x to a vertex y if and only if x is related to y.

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 converse relation. For the theoretical explanation see Category of relations.

Particular homogeneous relations

Some important particular binary relations on a given set X are:

• the empty relation E = ∅ ⊆ X×X,
• the universal relation U = X×X, and
• the identity relation I = { (x,x) : x∈X }.

For arbitrary elements x, y of X,

• xEy holds never,
• xUy holds always, and
• xIy holds if, and only if, x=y.

Properties

Some important properties that a binary relation R over a set X may have are:

• reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" (≥) is a reflexive relation but "greater than" (>) is not.
• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.
• coreflexive relation: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.
• quasi-reflexive: for all x, y in X, if xRy, then xRx and yRy.

The previous 4 alternatives are far from being exhaustive; e.g. the red relation y=x² from the above picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair (0,0), and (2,4), but not (2,2), respectively. The latter two facts also rule out quasi-reflexivity.

• symmetric: for all x and y in X it holds that if xRy then yRx. "Is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric; so is >, but vacuously (the condition in the definition is always false).
• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is both anti-symmetric and irreflexive. For example, > is asymmetric, but ≥ is not.

Again, the previous 3 alternatives are far from being exhaustive; as an example on the natural numbers, the relation xRy defined by x>2 is neither symmetric nor antisymmetric, let alone asymmetric.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric.
• connex: for all x and y in X it holds that xRy or yRx (or both). This property is sometimes called "total", which is distinct from the definitions of "total" given in the previous section.
• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomous relation, while the relation "divides" on natural numbers is not.
• right Euclidean (or just Euclidean): for all x, y and z in X, if xRy and xRz, then yRz. For example, equality is a Euclidean relation because if x=y and x=z, then y=z.
• left Euclidean: for all x, y and z in X, if yRx and zRx, then yRz.
• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. But it is not a serial relation on the positive integers, because there is no y in the positive integers such that 1>y. However, "is less than" is a serial relation on the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x.
• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relations on proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse > is not.
• well-founded: every nonempty subset S of X contains a minimal element with respect to R. Well-foundedness implies the descending chain condition (that is, no infinite chain ... x_n R ... R x₃ R x₂ R x₁ can exist). If the axiom of choice is assumed, both conditions are equivalent.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric, transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily being reflexive) is called a partial equivalence relation.

A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total (in the sense of connex) is called a total order, simple order, linear order, or a chain. A linear order where every nonempty subset has a least element is called a well-order.

Binary endorelations by property

	reflexivity	symmetry	transitivity	symbol	example
directed graph				→
undirected graph	irreflexive	symmetric
tournament	irreflexive	antisymmetric			pecking order
dependency	reflexive	symmetric
strict weak order	irreflexive	antisymmetric	yes	<
total preorder	reflexive		yes	≤
preorder	reflexive		yes	≤	preference
partial order	reflexive	antisymmetric	yes	≤	subset
partial equivalence		symmetric	yes
equivalence relation	reflexive	symmetric	yes	∼, ≅, ≈, ≡	equality
strict partial order	irreflexive	antisymmetric	yes	<	proper subset

The number of homogeneous relations

The number of distinct binary relations on an n-element set is 2ⁿ² (sequence A002416 in the OEIS):

Number of n-element binary relations of different types

n	all	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	65536	3994	4096	355	219	75	24	15
n	2n2		2n2−n			Σn k=0  k! S(n, k)	n!	Σn k=0  S(n, k)
OEIS	A002416	A006905	A053763	A000798	A001035	A000670	A000142	A000110

Notes:

• The number of irreflexive relations is the same as that of reflexive relations.
• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.
• The number of strict weak orders is the same as that of total preorders.
• The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
• The number of equivalence relations is the number of partitions, which is the Bell number.

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).

Examples of common homogeneous relations

• order relations, including strict orders:
  • greater than
  • greater than or equal to
  • less than
  • less than or equal to
  • divides (evenly)
  • is a subset of
• equivalence relations:
  • equality
  • is parallel to (for affine spaces)
  • is in bijection with
  • isomorphy
• tolerance relation, a reflexive and symmetric relation
  • dependency relation, a finite tolerance relation
  • independency relation, the complement of some dependency relation
• kinship relations

Other uses of correspondence

• In algebraic geometry, a correspondence is a binary relation or correspondence that is defined by a system of algebraic equations.
• In category theory, a correspondence from $C$ to $D$ is a functor $C^{\text{op}}\times D\to \mathbf {Set}$. It is the "opposite" of a profunctor.
• In von Neumann algebra theory, a correspondence is a synonym for a von Neumann algebra bimodule.
• In economics, a correspondence between two sets $A$ and $B$ is a map $f:A\to P(B)$ from the elements of the set $A$ to the power set of $B$. This is similar to a correspondence as defined in general mathematics (i.e., a relation,) except that the range is over sets instead of elements. However, there is usually the additional property that for all a in A, f(a) is not empty. In other words, each element in A maps to a non-empty subset of B; or in terms of a relation R as subset of A×B, R projects to A surjectively. A correspondence with this additional property is thought of as the generalization of a function, rather than as a special case of a relation, and is referred to in other contexts as a multivalued function.

An example of a correspondence in this sense is the best response correspondence in game theory, which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function. If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.

See also

• Abstract rewriting system
• Additive relation
• Confluence (term rewriting)
• Correspondence (algebraic geometry)
• Hasse diagram
• Incidence structure
• Logic of relatives
• Order theory
• Triadic relation

References

• Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.
• M. Kilp, U. Knauer, A.V. Mikhalev (2000) Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, ISBN 3-11-015248-7.
• Charles Saunders Pierce (1870) Description of a Notation for the Logic of Relatives from Google Books
• Gunther Schmidt (2010) Relational Mathematics Cambridge University Press ISBN 978-0-521-76268-7.

External links

• Media related to Binary relations at Wikimedia Commons
• Hazewinkel, Michiel, ed. (2001) [1994], "Binary relation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4