### Notation

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 a, b Î 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. (ii) Symmetric and antisymmetric relation: A relation R = R+, which is its own converse is called symmetric: a R b Û b R a 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 aÎ V otherwise R is antireflexive. (iv) Transitive relation: A relation R is transitive when aRb and bRc imply aRc, i.e. A R b L b R c Þ a R c for all a, b, c Î V (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. (vi) 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 (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

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.

• Subset of elements dominating an element a.

• Subset of elements dominated by an element a.

L(a) = {l|lÎ V; lp a}

• Subset of comparable elements.

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 ".