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International Journal of Combinatorics
Volume 2011 (2011), Article ID 893061, 10 pages
Research Article

Some More Results on IF Soft Rough Approximation Space

1Department of Mathematics, Tripura University, Suryamaninagar, Tripura 799130, India
2Department of Mathematics, Yazd University, Yazd 89195-741, Iran

Received 6 September 2011; Accepted 20 November 2011

Academic Editor: Toufik Mansour

Copyright © 2011 Sharmistha Bhattacharya (Halder) and Bijan Davvaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

1. Introduction

Data mining is a technique of extracting meaningful information from large and mostly un-organized 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. 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.

2. 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 form𝑈=𝑥,𝜇𝑢(𝑥),𝛾𝑢(𝑥)𝑥𝑋,(2.1) where the functions 𝜇𝑢𝑋[0,1] and 𝛾𝑢𝑋[0,1] denote, respectively, the degree of membership and the degree of nonmembership of each element 𝑥𝑋 to the set 𝑈, and 0𝜇𝑢(𝑥)+𝛾𝑢(𝑥)1 for each 𝑥𝑋. 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 𝑈𝑐 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:𝑈=𝑥,𝜇𝑢(𝑥),𝛾𝑢(𝑥)𝑥𝑋,𝑉=𝑥,𝜇𝑣(𝑥),𝛾𝑣(𝑥)𝑥𝑋.(2.2) Then,(1)𝑈𝑐={𝑥,𝛾𝑢(𝑥),𝜇𝑢(𝑥)𝑥𝑋},(2)𝑈𝑉={𝑥,𝜇𝑢(𝑥)𝜇𝑣(𝑥),𝛾𝑢(𝑥)𝛾𝑣(𝑥)𝑥𝑋},(3)𝑈𝑉={𝑥,𝜇𝑢(𝑥)𝜇𝑣(𝑥),𝛾𝑢(𝑥)𝛾𝑣(𝑥)𝑥𝑋},(4)0={𝑥,0,1𝑥𝑋},1={𝑥,1,0𝑥𝑋},(5)(𝑈𝑐)𝑐=𝑈,0𝑐=1,1𝑐=0.

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 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)={𝑥𝑅(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)={𝑥𝑅(𝑥)𝑋𝜙}. For an approximation space (𝑈,𝜃), by a rough approximation in (𝑈,𝜃) we mean a mapping Apr𝒫(𝑈)𝒫(𝑈)×𝒫(𝑈) defined by for every 𝑋𝒫(𝑈),𝑅Apr(𝑋)=(𝑋),𝑅(𝑋).(2.3) Given an approximation space (𝑈,𝑅), a pair (𝐴,𝐵)𝒫(𝑈)×𝒫(𝑈) is called a rough set in (𝑈,𝑅) if (𝐴,𝐵)=Apr(𝑋) for some 𝑋𝒫(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 𝑋𝑇 there is an associated set 𝐼𝑋𝑇 such that the following conditions are satisfied: for any 𝑋,𝑌𝑇, 𝐼(𝑋𝑌)=𝐼𝑋𝐼𝑌, 𝐼𝑋𝑋, 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 𝐴𝐸 and 𝐹𝐴𝑃(𝑈), the power set of 𝑈, is a set-valued mapping.

Let (𝑈,𝑅) be a Pawlak approximation space. For a fuzzy set 𝜇𝐹(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 by𝑅[𝑥](𝜇)(𝑥)=𝜇(𝑦)𝑦𝑅,[𝑥]𝑅(𝜇)(𝑥)=𝜇(𝑦)𝑦𝑅,(2.4) for all 𝑥𝑈. The operators 𝑅 and 𝑅 are called the lower and upper rough approximation operators on fuzzy sets. If 𝑅=𝑅 the fuzzy set 𝜇 is said to be definable, otherwise 𝜇 is called a rough fuzzy set.

A soft set (𝐹,𝐴) over 𝑈 is called a full soft set if 𝑎𝐴𝐹(𝑎)=𝑈. Let (𝐹,𝐴) be a full soft set over 𝑈, and let 𝒮=(𝑈,𝐸) be a soft approximation space. For a fuzzy set 𝜇𝐹(𝑈) the lower and upper soft rough approximations of 𝜇 with respect to 𝑆 are denoted by sap𝑆(𝜇) and sap𝑆(𝜇), respectively, which are fuzzy sets in 𝑈 given bysap𝑆(𝜇)(𝑥)={𝜇(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)},sap𝑆(𝜇)(𝑥)={𝜇(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)},(2.5) for all 𝑥𝑈. The operators sap𝑆(𝜇) and sap𝑆(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.

3. 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.

Definition 3.1. Let  =(𝐹,𝐴)be a full soft set over 𝑈 and 𝒮=(𝑈,𝐸) a soft approximation space. 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 sap𝑆(𝜇,𝛾) and sap𝑆(𝜇,𝛾) and are defined as follows:sap𝑆(𝜇,𝛾)(𝑥)=sup𝛾inf𝜇{𝜇(𝑦),𝛾(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)},sap𝑆(𝜇,𝛾)(𝑥)=inf𝜇sup𝛾{𝜇(𝑦),𝛾(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)},(3.1) for all 𝑥𝑈.

Example 3.2. Suppose that 𝑈={𝑑1,𝑑2,𝑑3,𝑑4,𝑑5,𝑑6,𝑑7} is the universe of the days of a week and the set of parameters are given by 𝐸={𝑡1,𝑡2,𝑡3,𝑡4,𝑡5,𝑡6}, where 𝑡𝑖(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(𝑡1)={𝑑1,𝑑2}, 𝐹(𝑡2)={𝑑3,𝑑4,𝑑5}, 𝐹(𝑡3)={𝑑6,𝑑7}, 𝐹(𝑡4)={𝑑1,𝑑2}, 𝐹(𝑡5)={𝑑5,𝑑6}, 𝐹(𝑡6)={𝑑3,𝑑4}.
Suppose that 𝜇,𝛾=0.6,0.3𝑑1,0.6,0.4𝑑2,0.6,0.2𝑑3,0.5,0.2𝑑4,0.5,0.5𝑑5,0.3,0.7𝑑6,0.3,0.2𝑑7.(3.2) Then, sap𝑆(𝜇,𝛾)=0.6,0.4𝑑1,0.6,0.4𝑑2,0.5,0.5𝑑3,0.5,0.5𝑑4,0.3,0.7𝑑5,0.3,0.7𝑑6,0.3,0.7𝑑7,sap𝑆(𝜇,𝛾)=0.6,0.4𝑑1,0.6,0.4𝑑2,0.5,0.5𝑑3,0.5,0.5𝑑4,0.3,0.7𝑑5,0.3,0.7𝑑6,0.3,0.7𝑑7.(3.3)

Table 1

Remark 3.3. (1) sap𝑆(𝜇,𝛾)/(𝜇,𝛾) which follows from the above example. But it completely is a part of the same object.
(2) If any object is of the form 0,1, then sap𝑆0,1=sap𝑆0,1=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 sap𝑆1,0 need not be 1,0, since there may exist many other elements whose membership value is less than 1, but if sap𝑆1,0=1,0, then no other object is in the same mapping 𝐹. Similarly, sap𝑆1,01,0.
(4) Let any object 𝑑 be of the form 0,0. Now, if sap𝑆0,0=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.

Remark 3.4. (1) If sap𝑆(𝜇,𝛾)(𝑥)=sap𝑆(𝜇,𝛾)(𝑥), then the IF soft rough approximation is said to be simply IF soft approximation.
(2) If for some of the object sap𝑆(𝜇,𝛾)(𝑥)=sap𝑆(𝜇,𝛾)(𝑥), then the IF soft rough approximation is said to be simply IF soft oscillating approximation.
(3) If for none of the object sap𝑆(𝜇,𝛾)(𝑥)=sap𝑆(𝜇,𝛾)(𝑥), then the IF soft rough approximation is said to be completely IF soft rough approximation. For this case we may consider two more definitions which are known as IF soft stable lower rough approximation and IF soft stable upper rough approximations and are denoted by sap𝑆(𝜇,𝛾) and sap𝑆(𝜇,𝛾)(𝑥).

Definition 3.5. The positive difference between sap𝑆(𝜇,𝛾)  and  sap𝑆(𝜇,𝛾)(𝑥) is denoted by 𝑂𝑆 and is said to oscillate in the approximation space, that is, 𝑂𝑆=|||sap𝑆(𝜇,𝛾)sap𝑆|||(𝜇,𝛾),(3.4) where “||” is required since otherwise the membership value of the difference may be negative.

Example 3.6. Consider the Example 3.2. Then, we have 𝑂𝑆=0,0𝑑1,0,0𝑑2,0,0𝑑3,0,0𝑑4,0,0𝑑5,0,0𝑑6,0,0𝑑7=𝑂.(3.5) Now, let us consider another case of Example 3.2. Suppose that 𝜇,𝛾=0.3,0.6𝑑1,0.2,0.5𝑑2,0.4,0.5𝑑3,0.3,0.5𝑑4,0.5,0.5𝑑5,0.2,0.7𝑑6,0.4,0.3𝑑7.(3.6) Then, sap𝑆(𝜇,𝛾)(𝑥)=0.2,0.5𝑑1,0.2,0.5𝑑2,0.3,0.5𝑑3,0.3,0.5𝑑4,0.2,0.7𝑑5,0.2,0.7𝑑6,0.2,0.7𝑑7,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.6𝑑1,0.3,0.6𝑑2,0.3,0.5𝑑3,0.3,0.5𝑑4,0.2,0.7𝑑5,0.2,0.7𝑑6,0.2,0.7𝑑7,𝑂𝑆=0.1,0.1𝑑1,0.1,0.1𝑑2,0,0𝑑3,0,0𝑑4,0,0𝑑5,0,0𝑑6,0,0𝑑7𝑂.(3.7)

Theorem 3.7. If 𝑂𝑆=0,0 then we obtain object IF soft approximation space.

Proof. If 𝑂𝑆=0,0, then the following two cases may arise:(1)all the object has the same nonzero value for sap𝑆 and sap𝑆. Hence, from Remark 3.4, we obtain an IF soft approximation space.(2)If sap𝑆=0,0=sap𝑆, then the conclusions may be drawn from Remark 3.3.

Theorem 3.8. (1)  𝑂𝑆 can never be 1,0 for any object.
(2)  𝑂𝑆 can never be 0,1 for any object.

Proof. (1) Suppose that 𝑂𝑆=1,0, then from the definition we have |sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾|=1,0, that is, sup𝛾inf𝜇{𝜇(𝑦),𝛾(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)}sup𝜇inf𝛾{𝜇(𝑦),𝛾(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)}=1,0.(3.8) Let 𝑎,𝑏𝑐,𝑑=1,0, that is, 𝑎𝑐=1, 𝑑𝑏=0, that is, 𝑑=𝑏. Since 𝑎/1, so 𝑎𝑐=1 gives 𝑎=1 and 𝑐=0. Since 𝑎=1, 𝑏=0 gives 𝑑=0, that is, sap𝑆𝜇,𝛾=1,0,sap𝑆𝜇,𝛾=0,0,(3.9) but if 1,0 and 0,0 are members of the same mapping 𝐹, then sap𝑆𝜇,𝛾=0,0, which is a contradiction. Hence, 𝑂𝑆1,0.
(2) can be proved similarly.

Remark 3.9. (1) If 𝑂𝑆=0,0 for all object then the approximation space is IF soft approximation space for all object.
(2) If 𝑂𝑆=0,0 for some object then the approximation space is If soft oscillating space.
(3) If 𝑂𝑆0,0 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.

Definition 3.10. An IF soft stable lower rough approximation (IFSSLRA) of 𝜇,𝛾 with respect to 𝑆 is denoted by sap𝑆𝜇𝛾=sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾(3.10) and an IF soft stable upper rough approximation by sap𝑆𝜇𝛾=sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾.(3.11)

Example 3.11. Let us consider Example 3.2. Then, sap𝑆𝜇,𝛾=sap𝑆𝜇,𝛾, that is, sap𝑆𝜇𝛾=sap𝑆𝜇𝛾=sap𝑆𝜇𝛾=sap𝑆𝜇𝛾.(3.12) Now, if we consider Example 3.6 then sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾. Therefore, sap𝑆𝜇,𝛾=0.2,0.6𝑑1,0.2,0.6𝑑2,0.3,0.5𝑑3,0.3,0.5𝑑4,0.2,0.7𝑑5,0.2,0.7𝑑6,0.2,0.7𝑑7.(3.13)

Theorem 3.12. Let =(𝐹,𝐴) be a full soft set over 𝑈, and let 𝒮=(𝑈,𝐸) be a soft approximation space. Then, we have: (1)sap𝑆𝜇,𝛾=𝑖𝑛𝑓{𝜇(𝑦),𝛾(𝑦)𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)} and sap𝑆𝜇,𝛾=𝑠𝑢𝑝{<𝜇(𝑦),𝛾(𝑦)>𝑎𝐴,{𝑥,𝑦}𝐹(𝑎)},(2)sap𝑆𝜇,𝛾𝜇,𝛾sap𝑆𝜇,𝛾, sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾, sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾,(3)sap𝑆0,1=0,1 for any object 𝑑0,1 in 𝜇,𝛾,(4)sap𝑆0,00,1 for any object 𝑑0,0 in 𝜇,𝛾,(5)sap𝑆𝜙=𝜙=sap𝑆𝜙,(6)sap𝑆𝑈=sap𝑆𝑈,(7)sap(𝜇,𝛾𝜈,𝛽)=sap𝜇,𝛾sap𝜈,𝛽,(8)𝜇,𝛾𝜈,𝛽, sap𝜇,𝛾sap𝜈,𝛽,(9)sap(𝜇,𝛾𝜈,𝛽)sap𝜇,𝛾sap𝜈,𝛽.

Proof. It is straightforward.

Remark 3.13. If sap𝑆𝜇,𝛾=𝜇,𝛾, then 𝜇,𝛾 is an IF soft open set. In Example 3.11, {𝑑4,𝑑6} are soft open objects, and their memberships are soft open members. Also, if sap𝑆𝜇,𝛾=𝜇,𝛾, then 𝜇,𝛾 is a closed set.

Remark 3.14. (1) Here 𝐵𝑆=sap𝑆𝜇,𝛾sap𝑆𝜇,𝛾 is the IFSR boundary region.
(a) If 𝐵𝑆=0,0 then the data is IF soft set.
If 𝐵𝑆0,1, then the data are IF soft rough set. In Example 3.11, the first 𝜇,𝛾 is IF soft set and is not IF soft rough set but the second one is IFSR set.
(b) 𝐵𝑆=0,0 if and only if 𝑂𝑆=0,0.
(2) Here 𝑁𝑆=𝐼sap𝑆𝜇,𝛾, where 𝐼 denotes that the value 1,0 for every object is the IFSR negative region.
(a) If 𝑁𝑆=0, then sap𝑆𝜇,𝛾=𝐼 if and only if 𝜇,𝛾=𝐼, by Remark 3.3.
(b) 𝑁𝑆=𝐼 if sap𝑆𝜇,𝛾=0 if and only if 𝜇,𝛾=𝑂, where 𝑂 is the value <0,0> for every object.
Now, let us take an example from [7].

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={𝑡𝑠,𝑡𝑡,𝑏,𝑟,𝑑,𝑒𝑏𝑒𝑏𝑟}, where 𝑡𝑠 implies short height, 𝑡𝑡 implies tall height, 𝑏 implies blond hair, 𝑟 implies red hair, 𝑑 implies dark hair, 𝑒𝑏 implies blue eyes, and 𝑒𝑏𝑟 implies brown eyes. Let us consider a soft set (𝐹,𝐸) describing the “attractive person”. Let us represent Table 2.
Let 𝐹(𝑡𝑠)={1,2,8}, 𝐹(𝑡𝑡)={3,4,5,6,7}, 𝐹(𝑏)={1,2,8}, 𝐹(𝑟)={3}, 𝐹(𝑑)={4,5,7}, and 𝐹(𝑒𝑏)={1,3,4,5,6}, 𝐹(𝑒𝑏𝑟)={2,7,8}.
Let us now consider the IF set of an attractive person as per our choice as 𝜇,𝛾=0.8,0.11,0.3,0.72,0.6,0.33,0.3,0.54,0.4,0.55,0.7,0.26,0.2,0.87,0.3,0.58,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.71,0.2,0.82,0.2,0.83,0.2,0.84,0.2,0.85,0.2,0.86,0.2,0.87,0.2,0.88,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.71,0.3,0.72,0.2,0.83,0.2,0.84,0.2,0.85,0.2,0.86,0.2,0.87,0.2,0.88,𝑂𝑆=0,01,0.1,0.12,0,03,0,04,0,05,0,06,0,07,0,08,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.71,0.2,0.82,0.2,0.83,0.2,0.84,0.2,0.85,0.2,0.86,0.2,0.87,0.2,0.88,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.71,0.3,0.72,0.2,0.83,0.2,0.84,0.2,0.85,0.2,0.86,0.2,0.87,0.2,0.88,𝐵𝑆=𝑂𝑆,𝑁𝑆=0.7,0.31,0.7,0.32,0.8,0.23,0.8,0.24,0.8,0.25,0.8,0.26,0.8,0.27,0.8,0.28.(3.14) Here 𝐵𝑆=𝑂𝑆 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].

Table 2

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 𝜇,𝛾=0.8,0.11,0.5,0.42,0.9,0.13,0.3,0.44,0.6,0.35,0.7,0.26,sap𝑆(𝜇,𝛾)(𝑥)=0.5,0.41,0.5,0.42,0.3,0.43,0.3,0.44,0.5,0.45,0.5,0.46,sap𝑆(𝜇,𝛾)(𝑥)=0.9,0.11,0.9,0.12,0.9,0.13,0.9,0.14,0.9,0.15,0.9,0.16,𝑂𝑆=0.4,0.31,0.4,0.32,0.6,0.33,0.6,0.34,0.4,0.35,0.4,0.36,sap𝑆(𝜇,𝛾)(𝑥)=0.3,0.41,0.3,0.42,0.3,0.43,0.3,0.44,0.3,0.45,0.3,0.46,sap𝑆(𝜇,𝛾)(𝑥)=0.9,0.11,0.9,0.12,0.9,0.13,0.9,0.14,0.9,0.15,0.9,0.16,𝐵𝑆=0.6,0.31,0.6,0.32,0.6,0.32,0.6,0.34,0.6,0.35,0.6,0.36.(3.15) Here 𝐵𝑆𝑂𝑆 which implies that the approximation space may not be stable and the IF set taken for the persons is not perfectly correct.

Table 3

4. Conclusion

The concepts of IF lower soft rough approximation and IF upper soft rough approximation space are introduced. In the most of the cases, IFSLRA is not stable for that we had introduced a new concept of IFSSLRA space. In some sense almost all concepts we are meeting in every day life are vague rather than precise. This gap between real world and traditional mathematics becomes smaller in recent year. In order to remove this gap, rough set, IF set, a soft set help a lot. The data mining and decision making processes may cross a new milestone after introduction of this new hybridized model. Further study will be needed to establish the utilities of the notions indicated in this paper.


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