Binary relation

In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set.

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 to every integer z that is a multiple of p. In this relation, for instance, the prime 2 is associated to -4, 0, 6, 10, but not with 1 or 9; and the prime 3 is associated with 0, 6, and 9, but not with 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. The all-important concept of function is defined as a special case of binary relation. Binary relations are also heavily used in computer science, especially within the relational model for databases.

A binary relation is a special case of a k-ary relation, that is, a set of k-tuples where the j^th component of each k-tuple is taken from the j^th domain X_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.

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) ∈ G 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, i.e. 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. 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, 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

Total or partial

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).
right-total or surjective: for all y in Y there exists an x in X such that xRy.

Functional, injective, surjective, bijective

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.
bijective: left-total, right-total and functional.

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.

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. 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.
asymmetric: for all x and y in X it holds that if xRy then not yRx. "Greater than" is an asymmetric relation, because of 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.
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. (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. 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 is a binary relation over X and Y, then the following is a binary relation over Y and X:

Converse: R ^-1, defined as R ^-1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to its converse if and only if it is symmetric. The converse of a surjective and injective function is called its inverse.

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.
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-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:

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 }. The order of R and S in the notation S o R, used here agrees with the standard notational order for composition of functions.

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. (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.

Examples of common binary relations

Equality
Greater than
Greater than or equal to
Less than
Less than or equal to
Divides (evenly)

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