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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

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 no other. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without 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. In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects The all-important concept of function is defined as a special kind of binary relation. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Binary relations are also heavily used in computer science, especially within the relational model for databases. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their The relational model for Database management is a Database model based on first-order predicate logic, first formulated and proposed in 1969 by Edgar A Computer Database is a structured collection of records or data that is stored in a computer system

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. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple 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. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously 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. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the

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

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. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. The sets X and Y are called the domain and codomain, respectively, of the relation, and G is called its graph. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

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. In Mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of

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}.

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. In Mathematics, the notion of restriction finds a general definition in the context of sheaves. In Mathematics, the composition of Binary relations is a concept of forming a new relation S o R from two given relations R and S, having as In Mathematics, the inverse relation of a Binary relation is the relation taken 'backwards' as in changing the relation 'child of' to 'parent of' The choice between the two definitions usually matters only in very formal contexts, like category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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: 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: for all y in Y there exists an x in X such that xRy. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every
functional (also called right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z. 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
injective: 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

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. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

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. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. For example, "greater than or equal to" is a reflexive relation but "greater than" is not.
irreflexive: for all x in X it holds that not xRx. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. "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. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that "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. In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if "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. Asymmetric often means simply not symmetric In this sense an asymmetric relation is a Binary relation which is not a Symmetric relation. "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. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b "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). In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that "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. In Mathematics, a Binary relation R over a set X is euclidean if it holds for all a, b, and c in
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. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously (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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" A relation which is reflexive, antisymmetric and transitive is called a partial order. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement A partial order which is total is called a total order or a linear order or a chain. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation A linear order in which every nonempty set has a least element is called a well-order. In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every

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 }. In Mathematics, the inverse relation of a Binary relation is the relation taken 'backwards' as in changing the relation 'child of' to 'parent of' A binary relation over a set is equal to its inverse if and only if it is symmetric. See also duality (order theory). In the mathematical area of Order theory, every Partially ordered set P gives rise to a dual (or opposite) partially ordered set which

If R is a binary relation over X, then each of the following are binary relations 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 $^\neq$ , defined as R $^\neq$ = R \ {(x, x) | x ∈ X} or the largest irreflexive relation over X contained in R. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity.
Transitive closure: R ⁺, defined as the smallest transitive relation over X containing R. In Mathematics, the transitive closure of a Binary relation R on a set X is the smallest Transitive relation on X This can be seen to be equal to the intersection of all transitive relations containing R.
Transitive reduction: R ^-, defined as the smallest relation having the same transitive closure as R has. In Mathematics, the transitive reduction of a Binary relation R on a set X is a minimal Relation R' on
Transitive-reflexive closure: R ^*, defined as R ^* = (R ⁺) ⁼.

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

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 o R (also denoted R o S), defined as S o R = { (x, z) | there exists y ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. In Mathematics, the composition of Binary relations is a concept of forming a new relation S o R from two given relations R and S, having as The order of R and S in the notation S o R, used here agrees with the standard notational order for composition of functions. In Mathematics, a composite function represents the application of one function to the results of another

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. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

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. In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order

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. In Mathematics, the notion of restriction finds a general definition in the context of sheaves.

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. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if Asymmetric often means simply not symmetric In this sense an asymmetric relation is a Binary relation which is not a Symmetric relation. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that 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 In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"

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. In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered 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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, especially in Order theory, an upper bound of a Subset S of some Partially ordered set ( P, &le 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 $\subseteq$ 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 $\subseteq_A$ . Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation $\in_A$ 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. In the Foundations of mathematics, Von Neumann–Bernays–Gödel set theory ( NBG) is an Axiomatic set theory that is a Conservative extension In the Foundation of mathematics, Kelley–Morse (KM or Morse–Kelley (MK set theory is a first order Axiomatic set theory that is closely In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously (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 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 which are also total preorders. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement 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. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In combinatorial Mathematics, the n th Bell number, named in honor of Eric Temple Bell, is the number of partitions of a set

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its own complement. 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 The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement). In Mathematics, a quadruple or quadruplet is an ''n''-tuple with n being 4 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

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. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics and Computer science, a dependency relation is a Binary relation that is finite symmetric, and reflexive.
Independency relation, a symmetric, irreflexive relation. In Mathematics and Computer science, a dependency relation is a Binary relation that is finite symmetric, and reflexive.

Dictionary

-noun

(set theory) A relation, such as "is less than" or "is the daughter of", that makes statements about pairs of objects, these statements being true or false depending on the objects.

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