Approximation Properties on a Set Based on Equivalence Relations and Dominance Relations

An equivalence relation on a set forms equivalence classes so that the concept of approximation is formed on that set (rough set). The concept of approximation on a set is developing very rapidly. Some papers replace the equivalence relation with other relations, one of which is the dominance relation. The symmetry property of the equivalence relation is replaced by the anti-symmetry property so that a dominance relation is formed. This paper reviews several papers related to approximation on a set $w.r.t$ equivalence and dominance relations by describing the approximation properties that hold in both relations in terms of the concept of 3 types of approximation on a set. This paper also provides the approximation properties that hold in the equivalence relation but do not hold in the dominance relations in terms of the concept of 3 types of approximation on a set. The main contribution of this paper is showing that the relationship between the concept of 3 types of approximation on a set $w.r.t$ the equivalence relation and the dominance relation.