In mathematics, a relation on a set may, or may not, hold between two given set members. For example, “is less than” is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denotes as 1<3) , and likewise between 3 and 4 (denoted as 3<4), but neither between 3 and 1 nor between 4 and 4. As another example, “is sister of” is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation “to a certain degree”, hence e.g. “has some resemblance to” cannot be a relation.

Formally, a relation R over a set X can be seen as a set of ordered pairs (x, y) of members of X.[1] The relation R holds between x and y if (x, y) is a member of R. For example, the relation “is less than” on the natural numbers is an infinite set Rless of pairs of natural numbers that contains both (1,3) and (3,4), but neither (3,1) nor (4,4). The relation “is a nontrivial divisor of” on the set of one-digit natural numbers is sufficiently small to be shown here: Rdiv = { (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) }; for example 2 is a nontrivial divisor of 8, but not vice versa, hence (2,8) ∈ Rdiv, but (8,2) ∉ Rdiv.

If R is a relation that holds for x and y one often writes xRy. For most common relations in mathematics, special symbols are introduced, like “<” for “is less than”, and “|” for “is a nontrivial divisor of”, and, most popular “=” for “is equal to”. For example, “1<3”, “1 is less than 3”, and “(1,3) ∈ Rless” mean all the same; some authors also write “(1,3) ∈ (<)”.

Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x, and irreflexive if xRx holds for no x. It is symmetric if xRy always implies yRx, and asymmetric if xRy implies that yRx is impossible. It is transivitve if xRy and yRz always implies xRz. For example, “is less than” is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric, “is sister of” is symmetric and transitive, but neither reflexive (e.g. Pierre Curie is not a sister of himself) nor asymmetric, while being irreflexive or not may be a matter of definition (is every woman a sister of herself?), “is ancestor of” is transitive, while “is parent of” is not. Mathematical theorems are known about combinations of relation properties, such as “A transitive relation is irreflexive if, and only if, it is asymmetric”.

Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is irreflexive, asymmetric, and transitive, an equivalence relation is a relation that is reflexive, symmetric, and transitive,[citation needed] a function is a relation that is right-unique and left-total (see below).[2]

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations.[3][4][5] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

The above concept of relation[note 1] has been generalized to admit relations between members of two different sets (heterogeneous relation, like “lies on” between the set of all points and that of all lines in geometry), relations between three or more sets (Finitary relation, like “person x lives in town y at time z”), and relations between classes[note 2] (like “is an element of” on the class of all sets, see Binary relation § Sets versus classes).

Illustration of an example relation on a set A = { a, b, c, d }. An arrow from x to y indicates that the relation holds between x and y. The relation is represented by the set { (a,a), (a,b), (a,d), (b,a), (b,d), (c,b), (d,c), (d,d) } of ordered pairs.

1 Definition
2 Properties of homogeneous relations
3 Properties of (heterogeneous) relations
4 Operations on homogeneous relations
5 Operations on (heterogeneous) relations
6 Examples
7 See also

1 Definition

Given sets X and Y, the Cartesian product X × Y is defined as {(x, y) | x ∈ X and y ∈ Y}, and its elements are called ordered pairs.

A binary relation R over sets X and Y is a subset of X × Y.[1][6] The set X is called the domain[1] or set of departure of R, and the set Y the codomain or set of destination of R. In order to specify the choices of the sets X and Y, some authors define a binary relation or correspondence as an ordered triple (X, Y, G), where G is a subset of X × Y called the graph of the binary relation. The statement (x, y) ∈ R reads “x is R-related to y” and is written in infix notation as xRy.[3][4] The domain of definition or active domain[1] of R is the set of all x such that xRy for at least one y. The codomain of definition, active codomain,[1] image or 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 of definition and its codomain of definition.[7][8][9]

When X = Y, a binary relation is called a homogeneous relation (or endorelation).[10][11] Otherwise it is a heterogeneous relation.[12][13][14]

In a binary relation, the order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.

2 Properties of homogeneous relations

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

Reflexive
for all x ∈ X, xRx. For example, ≥ is a reflexive relation but > is not.
Irreflexive (or strict)
for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not.
The previous 2 alternatives are not exhaustive; e.g., the red binary relation y = x2 given in the section § Special types of binary relations is neither irreflexive, nor reflexive, since it contains the pair (0, 0), but not (2, 2), respectively.

Symmetric
for all x, y ∈ X, if xRy then yRx. For example, “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, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).[15]
Asymmetric
for all x, y ∈ X, if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[16] For example, > is an asymmetric relation, but ≥ is not.
Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric nor antisymmetric, let alone asymmetric.

Transitive
for all x, y, z ∈ X, if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric.[17] For example, “is ancestor of” is a transitive relation, while “is parent of” is not.
Dense
for all x, y ∈ X such that xRy, there exists some z ∈ X such that xRz and zRy. This is used in dense orders.
Connected
for all x, y ∈ X, if x ≠ y then xRy or yRx. This property is sometimes called “total”, which is distinct from the definitions of “total” given in the section Relation (mathematics) § Properties of (heterogeneous) relations.
Strongly connected
for all x, y ∈ X, xRy or yRx. This property is sometimes called “total”, which is distinct from the definitions of “total” given in the section Relation (mathematics) § Properties of (heterogeneous) relations.
Trichotomous
for all x, y ∈ X, exactly one of xRy, yRx or x = y holds. For example, > is a trichotomous relation, while the relation “divides” over the natural numbers is not.[18]
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 … xnR…Rx3Rx2Rx1 can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.[19][20]
Preorder
A relation that is reflexive and transitive.
Total preorder (also, linear preorder or weak order)
A relation that is reflexive, transitive, and connected.
Partial order (also, order[citation needed])
A relation that is reflexive, antisymmetric, and transitive.
Strict partial order (also, strict order[citation needed])
A relation that is irreflexive, antisymmetric, and transitive.
Total order (also, linear order, simple order, or chain)
A relation that is reflexive, antisymmetric, transitive and connected.[21]
Strict total order (also, strict linear order, strict simple order, or strict chain)
A relation that is irreflexive, antisymmetric, transitive and connected.
Partial equivalence relation
A relation that is symmetric and transitive.
Equivalence relation
A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.

3 Properties of (heterogeneous) relations

Examples of four types of binary relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

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

Uniqueness properties:

Injective (also called left-unique)[22]
For all x, z ∈ X and all y ∈ Y, if xRy and zRy then x = z. For such a relation, {Y} is called a primary key of R.[1] For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).
Functional (also called right-unique,[22]right-definite[23] or univalent)
[5] For all x ∈ X and all y, z ∈ Y, if xRy and xRz then y = z. Such a binary relation is called a partial function. For such a relation, {X} is called a primary key of R.[1] For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).
One-to-one
Injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
One-to-many
Injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
Many-to-one
Functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
Many-to-many
Not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.
Totality properties (only definable if the domain X and codomain Y are specified):

Total (also called left-total)
For all x in X there exists a y in Y such that xRy. In other words, the domain of definition of R is equal to X. This property is different from the definition of connected (also called total by some authors)[citation needed] in the section Properties. Such a binary relation is called a multivalued function. For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate −1 to any real number), nor the black one (as it does not relate 2 to any real number).
Serial (or left-total)
for all x ∈ X, there exists some y ∈ X such that xRy. For example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that 1 > y.[24] However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y = x.

Surjective (also called right-total[22] or onto)
For all y in Y, there exists an x in X such that xRy. In other words, the codomain of definition of R is equal to Y. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor the black one (as it does not relate any real number to 2).
Uniqueness and totality properties (only definable if the domain X and codomain Y are specified):

A function
A binary relation that is functional and total. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
An injection
A function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not.
A surjection
A function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not.
A bijection
A function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

4 Operations on homogeneous relations

If R is a homogeneous relation over a set X then each of the following is a homogeneous 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.
All operations defined in the section § Operations on binary relations also apply to homogeneous relations.

Homogeneous relations by property
Reflexivity Symmetry Transitivity Connectedness Symbol Example
Directed graph →
Undirected graph Symmetric
Dependency Reflexive Symmetric
Tournament Irreflexive Antisymmetric Pecking order
Preorder Reflexive Yes ≤ Preference
Total preorder Reflexive Yes Yes ≤
Partial order Reflexive Antisymmetric Yes ≤ Subset
Strict partial order Irreflexive Antisymmetric Yes < Strict subset
Total order Reflexive Antisymmetric Yes Yes ≤ Alphabetical order
Strict total order Irreflexive Antisymmetric Yes Yes < Strict alphabetical order
Partial equivalence relation Symmetric Yes
Equivalence relation Reflexive Symmetric Yes ∼, ≡ Equality

5 Operations on (heterogeneous) relations

Union
If R and S are binary relations over sets X and Y then R ∪ S = {(x, y) | xRy or xSy} is the union relation of R and S over X and Y. The identity element is the empty relation. For example, ≤ is the union of < and =, and ≥ is the union of > and =.
Intersection
If R and S are binary relations over sets X and Y then R ∩ S = {(x, y) | xRy and xSy} is the intersection relation of R and S over X and Y. The identity element is the universal relation. For example, the relation “is divisible by 6” is the intersection of the relations “is divisible by 3” and “is divisible by 2”.
Composition
If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then S ∘ R = {(x, z) | there exists y ∈ Y such that xRy and ySz} (also denoted by R; S) is the composition relation of R and S over X and Z. 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”. For the former case, if x is the parent of y and y is the mother of z, then x is the maternal grandparent of z.
Converse
If R is a binary relation over sets X and Y then RT = {(y, x) | xRy} is the converse relation of R over Y and X. For example, = is the converse of itself, as is ≠, and < and > are each other’s converse, as are ≤ and ≥. A binary relation is equal to its converse if and only if it is symmetric.
Complement
If R is a binary relation over sets X and Y then R = {(x, y) | not xRy} (also denoted by R or ¬ R) is the complementary relation of R over X and Y. For example, = and ≠ are each other’s complement, as are ⊆ and ⊈, ⊇ and ⊉, and ∈ and ∉, and, for total orders, also < and ≥, and > and ≤. The complement of the converse relation RT is the converse of the complement: {\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}
Restriction
If R is a binary homogeneous relation over a set X and S is a subset of X then R|S = {(x, y) | xRy and x ∈ S and y ∈ S} is the restriction relation of R to S over X. If R is a binary relation over sets X and Y and if S is a subset of X then R|S = {(x, y) | xRy and x ∈ S} is the left-restriction relation of R to S over X and Y. If R is a binary relation over sets X and Y and if S is a subset of Y then R|S = {(x, y) | xRy and y ∈ S} is the right-restriction relation of R to S over X and Y. 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, then so too are its restrictions. 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.
A binary relation R over sets X and Y is said to be contained in a relation S over X and Y, written {\displaystyle R\subseteq S,}{\displaystyle R\subseteq S,} if R is a subset of S, that is, for all {\displaystyle x\in X}x\in X and {\displaystyle y\in Y,}{\displaystyle y\in Y,} if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called equal written R = S. If R is contained in S but S is not contained in R, then R is said to be smaller than S, written R ⊊ S. For example, on the rational numbers, the relation > is smaller than ≥, and equal to the composition > ∘ >.

6 Examples

Order relations, including strict orders:
Greater than
Greater than or equal to
Less than
Less than or equal to
Divides (evenly)
Subset of
Equivalence relations:
Equality
Parallel with (for affine spaces)
Is in bijection with
Isomorphic
Tolerance relation, a reflexive and symmetric relation:
Dependency relation, a finite tolerance relation
Independency relation, the complement of some dependency relation
Kinship relations

7 See also

Abstract rewriting system
Additive relation, a many-valued homomorphism between modules
Category of relations, a category having sets as objects and heterogeneous binary relations as morphisms
Confluence (term rewriting), discusses several unusual but fundamental properties of binary relations
Correspondence (algebraic geometry), a binary relation defined by algebraic equations
Hasse diagram, a graphic means to display an order relation
Incidence structure, a heterogeneous relation between set of points and lines
Logic of relatives, a theory of relations by Charles Sanders Peirce
Order theory, investigates properties of order relations

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