Binary relation

In mathematics, the concept of binary relation is exemplified by such ideas as "is greater than" and "is equal to" in arithmetic, or "is congruent to" in geometry, or "is an element of" or "is a subset of" in set theory. But also functions are a special case of binary relations. Put in lay terms, a binary relation is a statement about two objects that may be true or false depending on the choice of objects, for example, "4 is less than 5" is true, and the relation is "is less than".

Contents

1 Definition and examples

1.1 Definition
1.2 Remark
1.3 Example

2 Special relations

3 Relations over a set

4 Operations on binary relations

5 Related topics

Definition and examples

Definition

Formally, a binary relation over a set X and a set Y is an ordered triple R=(X, Y, G(R)) where G(R), called the graph of the relation R, is a subset of the Cartesian product X × Y. If (x,y) ∈ G(R) then we say that x is R-related to y and write xRy or R(x,y).

Remark

It is common practice to identify the relation with its graph, i.e. if R ⊆ X × Y we call R a relation over X,Y. The distinction becomes important when one asks if the relation is total or surjective, or when dealing with restrictions and composition of relations (in particular, functions).

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 ball is owned by John.

Note that 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".

It may also be thought of as a boolean valued binary function that takes as arguments an element x of X and an element y of Y and evaluates to true or false (indicating whether the ordered pair (x, y) is an element of the set which is the relation).

Special relations

Some important properties that binary relation R over X and Y may or may not have are:

total: for all x in X there exists a y in Y such that xRy (note that this is a different definition for total from the one in the next section).
functional: for all x in X, and y and z in Y it holds that if xRy and xRz then y = z.
surjective: for all y in Y there exists an x in X such that xRy.
injective: for all x and z in X and y in Y it holds that if xRy and zRy then x = z.

A binary relation that is functional is called a partial function; a binary relation that is both total and functional is called a 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 properties that binary relations over a set X may or may not have 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: 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 of x≥y and y≥x, then x=y.
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. "Is greater than or equal to" is an example of a total relation (note that this is a different definition for total from the one in the previous section).
trichotomous: for all x and y in X exactly one of xRy, yRx and x = y holds. "Is greater than" is an example of a trichotomous relation.
extendibility: 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.

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 the least element is called a well-order.

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

Operations on binary relations

If R,S ⊆ X × Y are binary relations, then each of the following are binary relations:

Converse: R^-1 ⊆ Y × X, defined as R^-1 = { (y, x) | (x, y) ∈ R }
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 and S have the same base set for the domain and range (i.e. R,S ⊆ X × X) then we have

Composition: R o S defined as R o S = { (x,z) : there exists y such that (x,y) ∈ R and (y,z) ∈ S }

If a binary relation is also a binary function injective and onto, the converse is called inverse of the function.