Binary relation

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 A² = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.

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 not with any 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 ⊆ A₁ × … × A_n, that is, a set of n-tuples where the j^th component of each n-tuple is taken from the j^th domain A_j of the relation.

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.

Contents 1 Formal definition 1.1 Is a relation more than its graph? 1.2 Example 2 Special types of binary relations 3 Relations over a set 4 Operations on binary relations 4.1 Complement 4.2 Restriction 5 Sets versus classes 6 The number of binary relations 7 Examples of common binary relations 8 See also 9 Notes 10 References

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, 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 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 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

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

R=({ball, car, doll, gun}, {John, Mary, Ian, 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

Uniqueness:

• left-unique (also called injective)^[1]: for all x and z in X and y in Y it holds that if xRy and zRy then x = z.
• right-unique (also called right-definite or functional)^[1]: 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.
• one-to-one (also written 1-to-1): left-unique and right-unique.

Totality:

• 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).
• right-total (also called surjective)^[1]: for all y in Y there exists an x in X such that xRy.
• A correspondence: a binary relation that is both left-total and right-total.

Uniqueness and Totality:

• A function: a relation that is right-unique and left-total.
• A bijection: a one-to-one correspondence; such a relation is a function and is said to be bijective.

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 classes of endorelations are widely studied in graph theory, where they're known as directed graphs.

The set of all binary relations B(X) on a set X is a semigroup with involution with the involution being the mapping of a relation to its inverse relation.

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. "Equal to and odd" is an example of a coreflexive relation.
• 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: 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 (and zRy). Equality is a Euclidean relation because if x=y and x=z, then y=z.
• 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.^[2] Every reflexive relation is serial.
• 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 > is not.

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.^[3] A linear order where every nonempty set has a least element is called a well-order. A relation that is symmetric, transitive, and serial 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 ⁺) ⁼, the smallest preorder containing R.
• Reflexive transitive symmetric closure: R ^≡, defined as the smallest equivalence relation over X containing 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 (or more ambiguously 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 if 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, 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.

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.

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 that 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 2ⁿ² (sequence A002416 in 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
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 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
dependency	Yes	Yes
weak order			Yes	≤
preorder	Yes		Yes	≤	preference
partial order	Yes	No	Yes	≤	subset
partial equivalence		Yes	Yes
equivalence relation	Yes	Yes	Yes	∼, ≅, ≈, ≡	equality
strict partial order	No	No	Yes	<	proper subset

See also

Notes

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.