Let V= (a, b, c, ….) be a set of elements and a binary relation defined on it by R.


Binary relation: A binary relation R in V is such that for any two elements

a, b V

a R b

For each binary relation R in V there exists a converse relation R+ in V such that

b R a

The concept of binary relation itself is too general for practical use. However, it can be restricted by certain relational properties.


Symmetric and antisymmetric relation: A relation R = R+, which is its own converse is called symmetric:

a R b b R a

and vice versa. If this property does not hold, the relation is called antisymmetric.


Reflexive and antireflexive relation: A relation R is reflexive when aRa for all

a V

otherwise R is antireflexive.


Transitive relation: A relation R is transitive when aRb and bRc imply aRc, i.e.

A R b L b R c a R c for all a, b, c V


Equivalence relation: A relation R defined on a set V is an equivalent relation when it is

reflexive, symmetric and transitive

The commonly used 'equality' relation (=) defined on the set of real numbers is an example of equivalent relation.


Strict partial order relation: A relation

a p b

is called a strict partial order when it satisfies the conditions:

  • a p b and b p a cannot hold simultaneously
  • the relation "a" is transitive


Partially ordered set

A set V (a, b, c,) is called a partially ordered set when a strict partial order relation"a" is defined on it. The fundamental properties of a partially ordered set are:

ap b and bp c implies ap c (a, b, c, V)

ap b and bp a cannot hold simultaneously

A partially ordered set is denoted by (A; p )


Ordered set:

A set V is called an ordered set if there are two relations "≈" and "a " defined on it and they satisfy the axioms of ordering:

for any two elements a, b, V, one and only one of the relations a ≈ b, ap b, b<a holds.

"≈" is an equivalence relation, and

"p " is a transitive relation.

In other words, an ordered set is a partially ordered set with additional equivalence relation defined on it, and where the conditions "neither ap b" nor bp a" and "a ≈ b" are equivalent.

L(a) = {l|l V; lp a}

C(a) = G(a) U L(a)

Note that G(a) L(a) = f .

Strict dominance.

An element a strictly dominates an element b , if

a p b and not (b p a)

It can also be said that "a is strictly better than b ", or that "b is strictly worse than a ".