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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 a 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 Rrelated to y and write xRy or R(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
Note that two different relations could have the same graph. For example: the relation
Neverthesis, R is often identified 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).
Special Relations Some important properties that binary relation R over X and Y may or may not have are:
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.
Some important properties that binary relations over a set X may or may not have are:
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.
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