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# Binary relation

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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 R-related 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

R=({ball, car, doll, gun}, {John, Mary, So, Venus}, {(ball,John), (doll,Mary), (car,Venus)}).
The pair (ball,John), denoted by ballRJohn 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 a owner. But the graphs of the two relations are the same.

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:

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.

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
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

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.

-- Function -- Partial order -- Total order -- Well-order -- Equivalence relation --

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