Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A² = 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. 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.

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.

Formal definition

A binary relation R between arbitrary sets (or classes) X (the set of departure) and Y (the set of destination or codomain) is specified by its graph G, which is a subset of the Cartesian product X × Y. The binary relation R itself is usually identified with its graph G, but some authors define it as an ordered triple (X, Y, G), which is otherwise referred to as a correspondence.^[1]

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 of the subset G of X × Y, i.e. R(x, y) equals to 1 (true), if (x, y) ∈ G, and 0 (false) otherwise.

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.

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.^[2][3][4]

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. To emphasize that X and Y can be different sets, some authors call these heterogeneous relations.^[6][7]

Uniqueness properties:

• injective (also called left-unique^[8]): 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^[9] or right-unique^[8] or right-definite^[10]): 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:^[8] 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^[8] 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.^[11] The green relation is bijective, but the red is not.

Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.^[12] In computer science, such a relation is also called a homogeneous (binary) relation.^[7][12][13] 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 converse relation. For the theoretical explanation see Category of relations.

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.^[14] 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).^[15]
• 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.^[16] 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.^[17]
• 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.^[18]
• 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.^[19] 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.^[20] 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

Particular binary 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.

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 }. 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 }.

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 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.

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.)^[21] With this definition one can for instance define a function relation between every set and its power set.

The number of binary relations

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

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 binary 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

See also

Notes

References

• 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