Binary relation

In mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of another set.

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, and no other. 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, 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, especially within the relational model for databases.

A binary relation is the special case n = 2 of an n-ary relation, that is, a set of n-tuples where the j^th component of each n-tuple is taken from the j^th domain X_j of the relation. An n-ary relation among elements of a single set is said to be homogeneous.

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 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 and codomain(target), respectively, of the relation, and G is called its graph.

The statement (x,y) ∈ R 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 set of pairs 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.

Is a relation more than its graph?

According to the definition above, two relations with the same graph may be different, if they differ in the sets X and Y. For example, if G = {(1,2),(1,3),(2,7)}, then (Z,Z, G), (R, N, G), and (N, R, G) are three distinct relations.

Some mathematicians do not consider the sets X and Y to be part of the relation, and therefore define a binary relation as being a subset of X×Y, that is, just the graph G. According to this view, the set of pairs {(1,2),(1,3),(2,7)} is a relation from any set that contains {1,2} to any set that contains {2,3,7}.

A special case of this difference in points of view applies to the notion of function. Most authors insist on distinguishing between a function's codomain(target) and its range. Thus, a single "rule" like mapping every real number x to x² can lead to distinct functions f:R→R and g:R→R⁺, depending as the images under that rule are understood to be reals or, more particularly, 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 will consider surjectivity—or being onto—as a property of functions, while the latter will see 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

Example: Suppose there are four objects: {ball, car, doll, gun} and four persons: {John, Mary, So, Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. No one owns the gun and So owns nothing. Then the binary relation "is owned by" is given as

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

Thus the first element of R is the set of objects, the second is the set of people, and the last element is a set of ordered pairs of the form (object, owner).

The pair (ball, John), denoted by _ballR_John means that the ball is owned by John.

Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball,John), (doll, Mary), (car, Venus)})

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".

Special types of binary relations

Some important classes of binary relations R over X and Y are listed below

left-total^[1]: for all x in X there exists a y in Y such that xRy (this property, although sometimes also referred to as total, is different from the definition of total in the next section).
surjective or right-total^[1]: for all y in Y there exists an x in X such that xRy.
functional (also called right-definite or right-unique^[1]): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z.
injective (or left-unique^[1]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z.
bijective: left-total, right-total, functional, and injective. A bijective relation is sometimes called a 1-to-1 correspondence.

A binary relation that is functional is called a partial function; a binary relation that is both left-total and functional is called a function.

A binary relation that is both functional and injective is sometimes called a 1-to-1 relation.

A binary relation that is both left-total and right-total is sometimes called a correspondence.

Relations over a set

If X = Y then we simply say that the binary relation is over X. Or it is an endorelation over X.

Some important classes of binary relations over a set X 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. "Greater than" is an example of an irreflexive relation.
coreflexive: for all x and y in X it holds that if xRy then x = y.
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 it holds that if xRy and yRx then x = y. "Greater than or equal to" is an antisymmetric relation, because if x≥y and y≥x, then x=y.
asymmetric: for all x and y in X it holds that if xRy then not yRx. "Greater than" is an asymmetric relation, because if x>y then not y>x.
transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. "Is an ancestor of" is a transitive relation, because if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z.
total (or linear): for all x and y in X it holds that xRy or yRx (or both). "Is greater than or equal to" is an example of a total relation (this definition for total is different from the one in the previous section).
trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. "Is greater than" is an example of a trichotomous relation.
Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.
extendable (or serial): for all x in X, there exists y in X such that xRy. "Is greater than" is an extendable relation on the integers. But it is not an extendable relation on the positive integers, because there is no y in the positive integers such that 1>y.
set-like: for every x in X, the class of all y such that yRx is a set. (This makes sense only if we allow relations on proper classes.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse <⁻¹ is not.

A relation which is reflexive, symmetric and transitive is called an equivalence relation. A relation which is reflexive, antisymmetric and transitive is called a partial order. A partial order which is total is called a total order or a linear order or a chain. A linear order in which every nonempty set has a least element is called a well-order.

A relation which is symmetric, transitive, and extendable is also reflexive.

Operations on binary relations

If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

Inverse or converse: R ⁻¹, defined as R ⁻¹ = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to its inverse if and only if it is symmetric. See also duality (order theory).

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 seen 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.
Transitive reduction: R ⁻, defined as a minimal relation having the same transitive closure as R.
Reflexive transitive closure: R *, defined as R * = (R ⁺) ⁼.

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

Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }.
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), 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.

Complement

If R is a binary relation over X and Y, then the following too:

The complement S is defined as x S y iff not x R y.

The complement of the inverse is the inverse of the complement.

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.

The complement of the inverse has these same properties.

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, {{#invoke:Binary_relation|Relations_over_a_set|trichotomous}}, a partial order, total order, strict weak order, {{#invoke:Strict_weak_order|Total_preorders|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.

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.

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 "set of all sets", which is not a set in the usual set theory. 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 which is a set.

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.)

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 number of binary relations

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

Number of n-element binary relations of different types
Elements	Any	Transitive	Reflexive	Symmetric	Preorder	Partial order	Total preorder	Total order	Equivalence relation
0	1	1	1	1	1	1	1	1	1
1	2	2	1	2	1	1	1	1	1
2	16	13	4	8	4	3	3	2	2
3	512	171	64	64	29	19	13	6	5
4	65,536	3,994	4,096	1,024	355	219	75	24	15
n	2^n²		2ⁿ⁽ⁿ⁻¹⁾	2^n(n+1)/2			∑ⁿ _k=0 k!S(n, k)	n!	∑ⁿ _k=0 S(n, k)
OEIS	A002416	A006905	A053763	A006125	A000798	A001035	A000670	A000142	A000110

Note that S(n, k) refers to Stirling numbers of the second kind.

Notes:

The number of irreflexive relations is the same as that of reflexive relations
The number of {{#invoke:Partially_ordered_set|Strict_and_non-strict_partial_orders|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 which are also total preorders. The number of preorders which 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, {{#invoke:Binary_relation|Complement|complement}}), except that for n = 0 the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, {{#invoke:Binary_relation|Operations on binary relations|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

dependency relation, a symmetric, reflexive relation.
independency relation, a symmetric, irreflexive relation.

Binary relations by property
	reflexive	symmetric	transitive	symbol	example
directed graph				→
undirected graph	No	Yes
tournament	No	No			pecking order
weak order			Yes	≤
preorder	Yes		Yes	≤	preference
partial order	Yes	= ^[2]	Yes	≤	subset
equivalence relation	Yes	Yes	Yes	∼, ≅, ≈, ≡	equality
strict partial order	No	No	Yes	<	proper subset

Notes

^ ^a ^b ^c ^d Klip, Knauer and Mikhalev: p. 3
^ antisymmetric

References

M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3110152487.