Some More Results on IF Soft Rough Approximation Space

Fuzzy sets, rough sets, and later on IF sets became useful mathematical tools for solving various decision making problems and data mining problems. Molodtsov introduced another concept soft set theory as a general frame work for reasoning about vague concepts. Since most of the data collected are either linguistic variable or consist of vague concepts so IF set and soft set help a lot in data mining problem. The aim of this paper is to introduce the concept of IF soft lower rough approximation and IF upper rough set approximation. Also, some properties of this set are studied, and also some problems of decision making are cited where this concept may help. Further research will be needed to apply this concept fully in the decision making and data mining problems.


Introduction
Data mining is a technique of extracting meaningful information from large and mostly unorganized data banks.Data mining is one of the areas in which rough set is widely used.Data mining is the process of automatically searching large volumes of data for patterns using tools such as classifications, association, rule mining, and clustering.The rough set theory is a well understood format framework for building data mining models in the form of logic rules on the bases of which it is possible to issue predictions that allow classifying new cases.
In general whenever data are collected they are linguistic variables.Not only this, the answers are not always in Yes/No form.So, in this case to deal with such type of data IF set is a very important tool.
Data are in most of the cases a relation between object and attribute.Soft set is an important tool to deal with such types of data.
So, throughout this paper a combined approach of soft set, IF set, rough set is studied.Further study is required to find the application of this concept in the field of data mining.

International Journal of Combinatorics
Zadeh in 1965 1 introduced the concept of fuzzy set.This set contains only a membership function lying between 0 and 1.But while collecting data many cases may be there where data are missing so IF sets are reqd which consists of both membership value and nonmembership value.Atanassov 2 introduced the concept of IF set.Atanassov named it intuitionistic fuzzy set.But nowadays a problem arose due to the already introduced concept of intuitionistic logic.Hence, instead of intuitionistic fuzzy set, throughout this paper we are using the nomenclature IF set.
Rough sets introduced by Pawlak 3 are also a very useful tool for data mining problems where vagueness is the key factor.Molodtsov 4 introduced the concept of soft set, and in 2009 Feng et al. 5 introduced a combined notion of fuzzy set, rough set, and soft set to deal with complex data which arises in the most social science problems.
In this paper, our aim is to introduce the concepts of IF soft lower and IF soft upper rough approximations which help a lot for sorting the vague data and tending towards decision.

Basic Definitions
In this section, some of the important required concepts necessary to go further through this paper are shown.
Let X be a nonempty set, and let I be the unit interval 0, 1 .According to 2 , an intuitionistic fuzzy set IFS for short U is an object having the form where the functions μ u : X → 0, 1 and γ u : X → 0, 1 denote, respectively, the degree of membership and the degree of nonmembership of each element x ∈ X to the set U, and 0 ≤ μ u x γ u x ≤ 1 for each x ∈ X.An intuitionistic fuzzy topology IFT for short on a nonempty set X is a family τ of IFS's in X containing 0 ∼ , 1 ∼ and closed under arbitrary infimum and finite supremum 6 .In this case, the pair X, τ is called an intuitionistic fuzzy topological space IFTS for short and each IFS in τ is known as an intuitionistic fuzzy open set IFOS for short .The compliment U c of an IFOS is called an intuitionistic fuzzy closed set IFCS for short .
Let X be a nonempty set and let IFS's U and V be in the following forms: Then, Let U be a finite nonempty set, called universe and R an equivalence relation on U, called indiscernibility relation.The pair U, R is called an approximation space.By R x we mean that International Journal of Combinatorics 3 the set of all y such that xRy, that is, R x x R is containing the element x.Let X be a subset of U. We want to characterize the set X with respect to R. According to Pawlak's paper 3 , the lower approximation of a set X with respect to R is the set of all objects, which surely belong to X, that is, R * X {x : R x ⊆ X}, and the upper approximation of X with respect to R is the set of all objects, which are partially belonging to X, that is, R * X {x : R x ∩ X / φ}.For an approximation space U, θ , by a rough approximation in U, θ we mean a mapping Apr : P U → P U × P U defined by for every X ∈ P U , Apr X R * X , R * X .

2.3
Given an approximation space U, R , a pair A, B ∈ P Apr X for some X ∈ P U .Fuzzy set is defined by employing the fuzzy membership function, whereas rough set is defined by approximations.The difference of the upper and the lower approximation is a boundary region.Any rough set has a nonempty boundary region whereas any crisp set has an empty boundary region.The lower approximation is called interior, and the upper approximation is called closure of the set.By using these concepts, we can make a topological space.
A set T is said to be a topological space if with every X ⊂ T there is an associated set IX ⊂ T such that the following conditions are satisfied: for any X, Y ⊂ T , I X ∩ Y IX ∩ IY , IX ⊂ X, IIX IX, and IT T .The operation I is called an interior operation.This topological space is written by T, I .
Let U be a universal set and let E be a set of parameters.According to 4 , a pair F, A is called a soft set over U, where A ⊆ E and F : A → P U , the power set of U, is a set-valued mapping.
Let U, R be a Pawlak approximation space.For a fuzzy set μ ∈ F U , the lower and upper rough approximations of μ in U, R are denoted by R μ and R μ , respectively, which are fuzzy sets defined by for all x ∈ U.The operators sap S μ and sap S μ are called the lower and upper soft rough approximation operators on fuzzy sets.If both the operators are the same then μ is said to be soft definable, otherwise μ is said to be soft rough fuzzy set.

On IF Soft Rough Approximations
In this section, we introduce the concept of IF soft rough approximation.Some of its properties are studied and examples are presented.The main focus of this paper is to show the scope of this newly introduced concept in the field of data mining and decision making. 2 If any object is of the form 0, 1 , then sap * S 0, 1 sap * S 0, 1 0, 1 , since 0 is the infimum of all members and 1 is the supremum of all non members.
3 If any object is of the form 1, 0 , then sap * S 1, 0 need not be 1, 0 , since there may exist many other elements whose membership value is less than 1, but if sap * S 1, 0 1, 0 , then no other object is in the same mapping F. Similarly, sap * S 1, 0 / 1, 0 .4 Let any object d be of the form 0, 0 .Now, if sap * S 0, 0 0, 0 , then also there does not exist any object in the same mapping with membership 0 but if other object exists with membership nonzero its nonmembership must be 0.
2 can be proved similarly.
Remark 3.9. 1 If O S 0, 0 for all object then the approximation space is IF soft approximation space for all object.
2 If O S 0, 0 for some object then the approximation space is If soft oscillating space.
3 If O S / 0, 0 for all objects then the approximation space is IF soft rough approximation space.
In such cases we need to define an IFSLR set approximation space which is stable; else decisions cannot be drawn for any particular object.

3.13
Theorem 3.12.Let E F, A be a full soft set over U, and let S U, E be a soft approximation space.Then, we have: Proof.It is straightforward.

Table 1 d 1
For an IF set μ, γ , the IF soft lower rough approximation and IF soft upper rough approximation with respect to the soft approximation space S are denoted by sap * Example 3.2.Suppose that U {d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , d 7 } is the universe of the days of a week and the set of parameters are given by E {t 1 , t 2 , t 3 , t 4 , t 5 , t 6 }, where t i i 1, . . ., 6 stands for hot, medium, cold, heavy rain, medium rainy, and not raining.Let us consider a soft set F, E describing the weather.Let us represent Table 1.Then, F t 1 {d 1 , d 2 }, F t 2 {d 3 , d 4 , d 5 }, F t 3 {d 6 , d 7 }, F t 4 {d 1 , d 2 }, F t 5 {d 5 , d 6 }, F t 6 {d 3 , d 4 }.which follows from the above example.But it completely is a part of the same object.
Definition 3.1.Let E F, A be a full soft set over U and S U, E a soft approximation space.γ inf μ μ y , γ y : ∃a ∈ A, x, y ⊆ F a , μ sup γ μ y , γ y : ∃a ∈ A, x, y ⊆ F a , 3.1 for all x ∈ U.
Definition 3.10.An IF soft stable lower rough approximation IFSSLRA of μ, γ with respect to S is denoted by

Table 2
, then μ, γ is an IF soft open set.In Example 3.11, {d 4 , d 6 } are soft open objects, and their memberships are soft open members.Also, if sap S μ, γ μ, γ , then μ, γ is a closed set.Example 3.15.Suppose that U {1, 2, 3, 4, 5, 6, 7, 8} is the universe consisting of eight persons and the set of parameters are given by E {ht s , ht t , h b , h r , h d , e b e br }, where ht s implies short height, ht t implies tall height, h b implies blond hair, h r implies red hair, h d implies dark hair, e b implies blue eyes, and e br implies brown eyes.Let us consider a soft set F, E describing the "attractive person".Let us represent Table 2.Here B S O S which implies that the approximation space is stable and the IF set taken for the persons is correct and of less error.Finally, we consider another example from 3 .Example 3.16.Suppose that U {1, 2, 3, 4, 5, 6} is the universe consisting of six persons and the set of parameters are given by E {H, M, T }, where H implies headache, M implies musclepain,and T implies temperature.Let us consider a soft set F, E describing the "flu infected person".Let us represent Table 3.Let F H {2, 3, 5}, F M {3, 4, 6}, and F T {1, 2, 3, 5, 6}.Let us now consider the IF set of a flu infected person as per our choice as