Home > Binary relation

1 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).

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.

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 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).

2 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
• 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.

3 Relations over a set

If X = Y then we simply say that the binary relation is 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.
• 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 either xRy or yRx. "Is greater than or equal to" is an example of a total relation.
• 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.
• extendability: 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 relationIn mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive symmetric and transitive i. if the relation is written as ~ it holds for all a b and c in X that # (Reflexivity) a ~ a # (Symmetry) if a ~ b then b ~ a # (Trans. 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. A linear order in which every nonempty set has the least element is called a well-orderIn mathematics, a well-order (or well-ordering on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order is then called a well-ordered set . For example.

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