A binary relation is a mathematical concept to do with "relations", such as "is greater than" and "is equal to" in arithmetic, or "is an element of" in set theory.
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 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 &sube 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)}).
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)}
Neverthesis, 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).
Table of contents |
2 Relations over a set 3 Operations on binary relations |
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 = zA 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.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 ; symmetric: for all x and z in X it holds that if xRz then zRx ; antisymmetric: for all x and z in X it holds that if xRz and zRx then x = z ; transitive: for all x, y and z in X it holds that if xRy and yRz then xRz ; trichotomous: for all x and y in X exactly one of xRy, yRx and x = y holds ; extendability: for all x in X, there exists y in X such that xRy
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 relation which is trichotomous is called a total order or a linear order.
Operations on binary relations
If R,S &sube X × Y are binary relations, then each of the following are binary relations:
- Inverse: R^{-1} &sube Y × X, defined as R^{-1} = { (y, x) | (x, y) &isin R }
- Union: R &cup S &sube X × Y, defined as R &cup S = {(x,y) | (x,y) &is in R or (x,y) &isin S }
- Intersection: R &cap S &sube X × Y, defined as R &cap S = {(x,y) | (x,y) &is in R and (x,y) &isin S }