Let V= (a, b, c, ….) be a set of elements and a binary relation defined on it by R.
(i) 
Binary relation: A binary relation R in V is such that for any two elements
For each binary relation R in V there exists a converse relation R^{+} in V such that
The concept of binary relation itself is too general for practical use. However, it can be restricted by certain relational properties. 
(ii) 
Symmetric and antisymmetric relation: A relation R = R^{+}, which is its own converse is called symmetric:
and vice versa. If this property does not hold, the relation is called antisymmetric. 
(iii) 
Reflexive and antireflexive relation: A relation R is reflexive when aRa for all
otherwise R is antireflexive. 
(iv) 
Transitive relation: A relation R is transitive when aRb and bRc imply aRc, i.e.

(v) 
Equivalence relation: A relation R defined on a set V is an equivalent relation when it is
The commonly used 'equality' relation (=) defined on the set of real numbers is an example of equivalent relation. 
(vi) 
Strict partial order relation: A relation
is called a strict partial order when it satisfies the conditions:

(vii) 
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, Î V) ap b and bp a cannot hold simultaneously A partially ordered set is denoted by (A; p ) 
(viii) 
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, Î 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) = {llÎ V; lp a}
C(a) = G(a) U L(a)
Note that G(a)Ç 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 ".