Binary relation

A binary relation is a mathematical concept which systematizes concepts as “… is equal to or higher than…” in Arithmétique, or “… is element of the whole…” in Set theory. It is a particular case of general relation or correspondence . One finds also this Concept in Graph theory.

Introduction

In an abstract way, a relation between two Ensemble S is a proposal which binds certain elements of the first whole with other elements of the second unit.

On a unit F made up girls and a unit G made up boys, for example, one could define a relation “Alice loves Bernard”, or another relation “Beatrice knows Paul”… One can thus see the relation as being wire connecting of the elements of two units.

In the case of a finished unit, one can then try to represent the relation by a diagram: if F = {Lucie, Béatrice, Delphine, Alice} and if G = {Bernard, Antoine, Paul, Charles}, the relation likes can be schematized by the following diagram:

One will be able to deplore the fact that Delphine does not love anybody, that Lucie has a generous heart and that Charles can only feel.

One can also try to thus make the list of the couples in relation. (for more convenience, one will preserve only the first two letters of the first name)

gr. = {(Lu, Be), (Lu, An), (Lu, Pa), (Be, An), (Al, Pa), (Al, Be)}

In mathematics, a “couple” is made of two elements put between brackets in a particular order. The relation is defined in first approach like a whole of couples, i.e. if two elements are connected between them, then the couple is an element of the unit relation . If one calls F the whole of the girls, and G the whole of the boys, then the whole of all the possible couples is called “Cartesian product of F by G ” and is noted F×G and the relation likes is then defined by the unit F , the unit G and a subset of F×G .

Formal definition

A binary relation $\ mathcal {R}$ of a unit E towards a unit F is defined by a part $\ mathcal {G}$ of E×F .

If $(X, there) \ in \ mathcal {G}$ one says that X is in relation to there and one notes it “ $x \ mathcal {R} y$ ”.

In the particular case where E = F one says that $\ mathcal {R}$ is a definite binary relation on E or in E .
If E = F×F , one speaks about ternary Relation interns on F .
Plus generally, if E = F ^{N - 1}, one will speak about relation '' N '' - surface on F .

One will notice that it is necessary, in a binary relation, to specify the unit E (called starting together), the unit F (called together of arrival ) AND the part $\ mathcal {G}$ of $E \ times F$ called the graph of the relation.

A binary relation can be regarded as a function of E×F to value as a whole { Vrai , Faux }, and which with a couple ( X , there ) associates Vrai if X is in relation to there and Faux if not (indicating if the couple ( X , there ) is an element of the graph of the relation or not).

Composition and inversion

Composition
If $\ mathcal {R}$ is a relation of E in F and $\ mathcal {S}$ of F in G , one can define a relation $\ mathcal {S} \ circ \ mathcal {R}$ of E in G by:

$\ mathcal {G} _ {\ mathcal {S} \ circ \ mathcal {R}} = \ left \ {(X, there) \ in E \ times G \, | \, \ exists Z \ in F/\, (X, Z) \ in R \ and (Z, there) \ in S \ right \} \,$

Notation: if $\ mathcal {R}$ is a relation on a Ensemble E and N a Entier naturalness, one notes $\ mathcal {R} ^n$ the composition of $\ mathcal {R}$ with itself N time, with convention that $\ mathcal {R} ^0$ indicates the relation of equality on E .

Inversion
If $\ mathcal {R}$ is a relation of E on F , one can define a relation of F on E known as relation opposite or reciprocal , by:

$\ mathcal {G} _ {\ mathcal {R} ^ {- 1}} = \ left \ {(X, there) \ in F \ times E \, | \, (there, X) \ in \ mathcal {G} _ {\ mathcal {R}} \ right \} \,$ .

Examples:

“smaller than” and “larger than” is relations opposite one of the other.
“likes” and “by” are also opposite one of the other is liked.

Functional relation

When, for any element X of E , X is in relation only with 0 or 1 element there of F , it is said that the relation is functional . It is a particular case of function . In formal language, the preceding property is written:

$\ forall X \ in E, \ forall there \ in F, \ forall Z \ in F, (X, there) \ in \ mathcal {G} _ {\ mathcal {R}} \ and (X, Z) \ in \ mathcal {G} _ {\ mathcal {R}} \ Rightarrow (there = Z) \,$

For more precise details, to see the article “mathematical Function”.
Important example:

the diagonal of E is defined by:

$\ Delta_E = \ left \ {(X, X) \, | \, X \ in E \ right \} \,$ .

It is the graph of the relation of equality on E , noted “=_E”, or “=” in the absence of ambiguity on the unit concerned.
This relation is also a function, the identity of E , noted “ Id_E ”.

Relation on (or in) a unit

If E = F , one will speak about relation on (or in) E .

Properties related to reflexivity

Reflexive relation

The relation $\ mathcal {R}$ on E is reflexive if any element of E is in relation to itself, i.e. if:

$\ forall X \ in E, X \ mathcal {R} X \,$

A relation is thus reflexive if its graph contains the diagonal of E , i.e. if:

$\ Delta_E \ subseteq \ mathcal {G} _ {\ mathcal {R}} \,$
In other words, the intersection of the graph of the relation with the diagonal of E is equal to this diagonal.
Examples:

the relation of inclusion between units is reflexive: any unit is included in itself;

in a whole of numbers, the relation “is a divider of” is reflexive: any number is its clean Diviseur ;

in a whole of people, the relation “is same family that”…

is reflexive
The reflexive fence , noted “ $\ mathcal {R} ^ {refl} \,$ ”, of a relation $\ mathcal {R}$ on a unit E is the relation on E whose graph is the union of that of $\ mathcal {R}$ and of the diagonal of E :

$\ mathcal {G} _ {\ mathcal {R} ^ {refl}} = \ mathcal {G} _ {\ mathcal {R}} \ cup \ Delta_E \,$

Irréflexive relation

The relation $\ mathcal {R}$ on E is irréflexive if any element of E is not in relation to itself, i.e. if:

$\ forall X \ in E, X \ not \! \, \ mathcal {R} X \,$

A relation is thus irréflexive if its graph east disjoins diagonal of E , i.e. if:

$\ Delta_E \ course \ mathcal {G} _ {\ mathcal {R}} = \ empty \,$
The intersection of the graph of the relation with the diagonal of E is thus reduced to the empty set.
Examples:

the strict inequality on the whole is an example of irréflexive relation: no entirety is strictly lower than itself;

in a whole of people, the relation “is child of” is irréflexive: nobody is his own child;

in a Polyhedral , the relation “has one and only one common side with” is a irréflexive relation between its faces: no face has that only one common side with itself…

A relation on a whole of at least two elements can of course not be neither reflexive , nor irréflexive , it is enough that an element either in relation to him even and the other not.
The only at the same time reflexive and irréflexives relations are the relations whose graph is empty.

Properties related to symmetry

Symmetrical relation

The relation $\ mathcal {R}$ on E is symmetrical if when a first element of E is in relation to a second element of E , the second element is him also in relation to the first, i.e. if:

$\ forall (X, there) \ in E^2, (X \ mathcal {R} there) \ Rightarrow (there \ mathcal {R} X) \,$

A relation is thus symmetrical if its graph merges with that of its opposite relation, i.e. if:

$\ mathcal {G} _ {\ mathcal {R}} = \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} \,$
or:

$\ mathcal {G} _ {\ mathcal {R}} \ course \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} = \ mathcal {G} _ {\ mathcal {R}} \,$ .

Examples:

in a whole of people, the relation “is same family that” is symmetrical;

in a polyhedron, the relation “has one and only one common side with” is a symmetrical relation between its faces: if a face has a common side with another face, the latter has the same common side with the first face;

among the natural whole , the relation “forms an even product with” is symmetrical, because the multiplication of the entireties is commutative .

The symmetrical fence , noted “ $\ mathcal {R} ^ {sym} \,$ ”, of a relation $\ mathcal {R}$ on a unit E is the relation on E whose graph is the union of that of $\ mathcal {R}$ and of its reciprocal (or opposite):

$\ mathcal {G} _ {\ mathcal {R} ^ {sym}} = \ mathcal {G} _ {\ mathcal {R}} \ cup \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} \,$

This symmetrical fence is besides universal among the symmetrical relations containing $\ mathcal {R}$ (what here, without going into categorical considerations , means that it is smallest!).

Antisymmetric relation

The relation $\ mathcal {R}$ on E is antisymmetric or slightly antisymmetric if when two elements of E are in mutual relation, they in fact are confused, i.e. if:

$\ forall (X, there) \ in E^2, (X \ mathcal {R} there) \ wedge (there \ mathcal {R} X) \ Rightarrow (X = there) \,$

A relation is thus slightly antisymmetric if the intersection of its graph with that of its reciprocal is included in the diagonal of E , i.e. if:

$\ mathcal {G} _ {\ mathcal {R}} \ course \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} \ subseteq \ Delta_E \,$ .

Examples:

relations “larger than” and “smaller than” on the natural whole or on the real .

the relation “divides” in the whole of the natural entireties

When a relation is at the same time antisymmetric and irréflexive, it is said sometimes that it is strongly antisymmetric . One can then simplify the definition.
The relation $\ mathcal {R}$ on E is strongly antisymmetric , i.e. antisymmetric and irréflexive, if when a first element of E is in relation to a second element of E , the second element is not in relation to the first, i.e. if:

$\ forall (X, there) \ in E^2, (X \ mathcal {R} there) \ Rightarrow (there \ not \! \, \ mathcal {R} X) \,$

A relation is thus strongly antisymmetric if the intersection of its graph with that of its reciprocal is empty, i.e. if:

$\ mathcal {G} _ {\ mathcal {R}} \ course \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} = \ empty \,$ .

Examples:

the relations of a strict nature, as the relation “is strictly larger than” on the entireties or realities, or the relation of strict inclusion are strongly antisymmetric.

in a whole of people, the relation “is child of” is asymmetrical: nobody is his own child, nor a fortiori the child of his children…

For a relation which one knows in addition that it is irréflexive, the strong antisymetry and the antisymetry are equivalent, and thus most of the time one speaks simply about antisymetry.
The only symmetrical relations and strongly antisymmetric are the empty relations. On the other hand the equality on any unit is an at the same time symmetrical and antisymmetric relation.
A relation can not be neither symmetrical nor antisymmetric, such as for example the relation of divisibility on the relative whole .

Transitivity

The relation $\ mathcal {R}$ on E is transitive if when a first element of E is in relation to a second element itself in relation to a third, the first element is also in relation to the third, i.e. if:

$\ forall (X, there, Z) \ in E^3, (X \ mathcal {R} there) \ wedge (there \ mathcal {R} Z) \ Rightarrow (X \ mathcal {R} Z) \,$

A relation $\ mathcal {R}$ is thus transitive if its graph contains that of its made up with itself, i.e. if:

$\ mathcal {G} _ {\ mathcal {R} \ circ \ mathcal {R}} \ subseteq \ mathcal {G} _ {\ mathcal {R}} \,$

Example:

the relation $\ leq$ on the natural whole is transitive.

One calls transitive fence $\ mathcal {R}$ the relation

$\ bigcup_ {N \ geq 1} \ mathcal {R} ^n$
it is universal among the transitive relations containing $\ mathcal {R}$ . It is noted “ $\ mathcal {R} ^ {trans}$ ”.

Total relation

The relation $\ mathcal {R}$ on E is total so for any pair of elements of E , it institutes at least a bond between the two elements considered, i.e. if:

$\ forall (X, there) \ in E^2, (X \ mathcal {R} there) \ vee (there \ mathcal {R} X) \,$

The relation is thus total if the union of its graph with that of its reciprocal is equal to the Cartesian square of E , i.e. if:

$\ mathcal {G} _ {\ mathcal {R}} \ cup \ mathcal {G} _ {\ mathcal {R} ^ {- 1}} = E^2 \,$

Example: the relation $\ leq$ on the whole of realities is a total relation.
Counterexample: the “divided” relation on the whole of the natural entireties is not total.

Relation of equivalence

A relation of equivalence is a reflexive, transitive and symmetrical relation. The simplest example of relation of equivalence is the equality. Into arithmetic the relation of Congruence modulo a given entirety is a relation of equivalence.
For more information to see the article “Relation of equivalence”.

Relation of order

A relation of order is a reflexive, transitive and antisymmetric relation.
If the relation is total then one says that the order is total . It is the case of the relation “larger than” on the natural whole . All the elements are not inevitably comparable by a relation of order; for example two natural entireties are not inevitably comparable by divisibility. It is said whereas divisibility is a partial order on NR .
More details in the article “Relation of order”.

Quite founded relation

See also: quite founded Relation

Examples

the relation of membership on $E \ times \ mathcal {P} (E)$

the relation of inclusion on $\ mathcal {P} (E)$ (relation of order)

the relation inferior or superior on $\ mathbb {R}$ (relations of orders)

the relation divides on $\ mathbb {NR}$ (relation of order)

the relation of equality (congruencielle or not) on E (relation of equivalence)

Many binary relations on finished units

Let us consider a unit E finished of cardinal N and a unit F finished of cardinal p . We can easily show that there are as many binary relations of E on F than of applications of E×F in {0, 1}, which gives 2^Np relations.
In particular, if E = F , one finds $2^ {n^2} \,$ binary relations on E , of which

$2^ {N (N - 1)} \,$ reflexive relations

$2^ {N (N + 1) /2} \,$ symmetrical relations

For the number of transitive relations, it still does not have currently a “closed” formula there

The number of relations of equivalence is equal to the number of partitions of a unit, i.e. the Nombre of Beautiful.

See too

Relation of equivalence

Relation of order

function

Random links: Ã“xido nitroso | Guggenmusik | Cléry-on-nap | Algorithm of Dekker | Anne-Louise Benedicte of Bourbon-Cop | Autotrophic cell | LambdaMOO