A STUDY ON SOFT ROUGH MINIMAL CONTINUOUS AND SOFT ROUGH MAXIMAL CONTINUOUS

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INTRODUCTION
Rough set was introduced by Pawlak [1] for dealing with vagueness and granularity in information systems. This theory deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximation. The reference space in rough set theory is the approximation space whose topology is generated by the equivalence classes of R. Rough set theory was proposed as a new approach to processing of incomplete data. One of the aims of the rough set theory is a description of imprecise concepts.
Suppose we are given a finite non empty set U of objects called universe. Each object of U is characterized by a description, for example a set of attributes values. In rough set theory an equivalence relation (reflexive, symmetric, and transitive relation) on the universe of objects is defined based on their attribute values. In particular, this equivalence relation constructed based on the equality relation on attribute values.
In the year 1999, Russian specialist Molodtsov [2], started the idea on soft sets as another scientific instrument to manage vulnerabilities while demonstrating issues in building material sciences, software engineering, financial aspects, sociologies and restorative sciences as a general mathematical tool for dealing with uncertain objects. To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems. While probability theory, fuzzy set theory [4], rough set theory [1,5], and other mathematical tools are well known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out by Molodtsov in [2,6]. The concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set theory is free from the difficulties affecting existing methods. Feng Feng et.,al [3] used soft set theory to generalize Pawlak rough set model. Based on the novel granulation structures they introduced soft approximation spaces, soft rough approximation and soft rough sets and studied some of the properties of soft approximation spaces and the same properties of Pawlak approximation space.
Presently, works on soft set theory are progressing rapidly. Maji et al. [7] defined several operations on soft sets and made a theoretical study on the theory of soft sets. Aktas and Cagman [8] compared soft sets to the related concepts of rough sets. M.Lellies Thivagar et.,al [9] introduced rough topology and Muhammad Shabir, et.,al [10] introduced soft topological spaces. i) The lower approximation space of a subset is defined as: The upper approximation space of a subset is defined as: The boundary region of X with respect to R is defined as: The set X is said to be rough with respect to R, if ( ) ≠ ( ), i. e., ( ) ≠ ∅ Definition 2.3: [10] Let be the collection of soft sets over X, then is said to be a soft topology on X if 1) ∅ ̃ belong to 2) The union of any number of soft sets in belong to 3) The intersection of any two soft sets in belongs to .
The triplet ( , , ) is called a soft topological space over X. ii) The union of elements of any sub collection of is in iii) The intersection of the elements of any finite sub collection of is in forms a topology on U called as a rough topology on U with respect to X. We call ( , , ) as a rough topological space.
Definition 2.5: [11] Let = ( , ) be a Soft set over U. The pair = ( , ) is called a Soft approximation space. Based on , we define the following two operators: assigning to every subset X ⊆ U two sets ( ) and ( ) called the lower and upper soft rough approximations of X in S , respectively. If ( ) = ( ), X is said to be soft definable; Otherwise X is called a soft rough set.
Clearly, ( ) and ( ) can be expressed equivalently as:     Proof: It follows from definitions.