# New Topological Approaches to Generalized Soft Rough Approximations with Medical Applications

Authors: Mostafa K. El-Bably; Muhammad I. Ali; El-Sayed A. Abo-Tabl
Journal: Journal of Mathematics
 (2021)

---

## Abstract

There are many approaches to deal with vagueness and ambiguity including soft sets and rough sets. Feng et al. initiated the concept of possible hybridization of soft sets and rough sets. They introduced the concept of soft rough sets, in which parameterized subsets of a universe set serve as the building blocks for lower and upper approximations of a subset. Topological notions play a vital role in rough sets and soft rough sets. So, the basic objectives of the current work are as follows: first, we find answers to some very important questions, such as how to determine the probability that a subset of the universe is definable. Some more similar questions are answered in rough sets and their extensions. Secondly, we enhance soft rough sets from topological perspective and introduce topological soft rough sets. We explore some of their properties to improve existing techniques. A comparison has been made with some existing studies to show that accuracy measure of proposed technique shows an improvement. Proposed technique has been employed in decision-making problem for diagnosing heart failure. For this two algorithms have been given.


---

## Body

## 1. Introduction

Mathematical modeling for the vagueness and uncertainty of data has many different methods, for instance, rough set theory [1], fuzzy set theory [2], soft set theory [3], and topology [4]. Pawlak [1] introduced the classical rough sets model in the early eighties to study vagueness of data, which originate from daily life situations. The key of this methodology is an equivalence relation which is constructed from the data of an information system. In general, it is very difficult to find an equivalence relation in such data. Therefore application of this technique is very limited. Therefore authors relaxed the condition of equivalence relation by some more general relations such as similarity relations (reflexive and symmetric) [5, 6], preorder relations (reflexive and transitive) [7], reflexive relations [8], general binary relations [8–14], topological approaches [15, 16], and coverings [17–19].Soft set theory is another mathematical model to deal with uncertainty, when data is collected from real-life situations. This concept was introduced by Molodtsove [3]. This theory has applications in many fields, for instance, game theory, operations research, integration of Riemann, and measurement theory [3]. Recently, scientists and researchers have shown their inclination to the idea of soft sets to apply it in numerous areas. For more information about this theory and its applications, we refer the reader to the references (soft set theoretical concepts [20, 21], soft sets and soft topological spaces [22–24], soft rough sets and their applications [25–28], and medical applications of soft sets and their extensions [29–35]).In rough set theory [1], basic requirement is to have an equivalence relation among the elements of the set under consideration. But in daily life situations it is not easy to find such an equivalence relation. Perhaps this limitation is associated with rough set theory due to the lack of parameterization tools. The idea of soft rough sets was initiated and studied by Feng et al. in [24] which are very useful in intelligent systems. The concept of the lower (resp., upper) approximation of this theory is particularly useful to extract knowledge hidden in an information system. Decision-making has a crucial part in our daily life, and this method produces the best alternate among dissimilar selections. Chen et al. [34] proposed the choice values of objects in a soft set and considered how to use this notion to address decision-making problems. In [35], Roy and Maji generalized this method for new decision-making problems. There are several subsequent advances after Maji et al.’s work, such as the uni-int decision-making using soft set theory [36]; Jha et al.’s [37] neutrosophic soft set notion in decision-making problems for stock trending analysis, and medical applications [38].Feng et al. replaced the classes of the equivalence relation by parameterizing subsets of a subset of the universe to define its approximations. In fact, Feng et al. have succeeded in proving that Pawlak’s rough set model is a specialization of the soft rough set as shown by Theorem 4.4 and Theorem 4.5 in [24]. It is worth noting that the concept of full soft sets deserves special attention for both theoretical and practical reasons. Theoretically, some typical properties of Pawlak’s rough sets hold for soft rough sets if and only if the underlying soft set in the soft approximation space is full. Pragmatically, it is justifiable to consider full soft sets in real-life applications. In fact, if a soft set is not full, it means that the available parameters are insufficient, and there exists at least one object which cannot be described by any of the parameters in the given soft set. With the help of soft rough approximations, some equivalent characterizations of full soft sets were given in [24]. In this paper, a new technique is given to define lower and upper approximations of a set with the help of topology generated by the given soft set; this is known mathematically as the notion of topological soft rough sets TSR−sets.The main contribution in the existing work is to present another model for soft rough sets without any restrictions and satisfy the characteristics of Pawlak’s rough sets. In other words, we propose a method for modifying soft rough sets from a topological point of view, so a new link between soft sets and general topology is proposed.First, we discuss the concept of the topology of all definable sets in rough set theory [1] and in soft rough sets [24]. Accordingly, we able to respond with the next very interesting questions:What is the probability that a subset of the universeU may be a definable set?What is the probability that the lower approximation of a nonempty subset ofU may be an empty set?What is the probability that the upper approximation of a proper subset ofU may be U?Secondly, a general topology is generated from the soft set to modify and generalize soft rough sets proposed in [24]. The suggested techniques extend the way for more applications of the general topology in soft rough sets theory. In fact, we use the image of parameters as a subbasis for a unique topology generated by a soft set, denoted by TSR. New generalized soft rough approximations, called “topological soft rough approximations” (briefly, TSR-approximations), are defined. It is shown that accuracy of proposed technique is higher than soft rough sets, due to reduction of boundary region. The importance of proposed approximations is clear from the fact that these not only reduce the boundary region but also satisfy basic properties similar to rough sets. Several comparisons among the present method and the preceding one [34] are obtained. Numerous examples are suggested to exemplify the relations between the topological soft rough sets and soft rough sets.Finally, some medical applications in the medical diagnosis of heart failure problems [39] are introduced. These applications illustrate the importance of the suggested methods in real-life problems. In fact, we apply a topological reduction for data set comprising the effect of five indications for twenty patients with heart failure disease. Accordingly, we can identify the core factors of the heart failure diagnosis. A comparison has been made between proposed technique and some already existing in the literature which shows the usefulness of proposed technique. Two algorithms are given based on proposed technique. The proposed algorithms are tested on hypothetical data for the purpose of comparison with already existing methods.


## 2. Basic Concepts

The current section is devoted to present some elementary definitions and consequences that are applied through paper are mentioned.


### 2.1. Topological Space

A topology [4] of a set U is defined by the collection τ of subsets of U which fulfills the following three axioms:(T1) ϕ, U∈τ.(T2) A finite intersection of subsets of τ is a member in τ.(T3) An arbitrary union of subsets of τ is a member in τ.We call a pairU,τ “topological space” or “space” and the members of U “points” of τ, and the subsets of U that belong to τ are said to be “open” sets and the complements of the open sets are called “closed” sets in the space. The collection of all closed sets denotes τc.An interiorintAresp.,closureclA of a subset A is given by a union of all open sets contained in Aresp., intersection of all closed sets that containA, formally:(1)intA=∪V∈τ:V⊆A,clA=∩Z∈τc:A⊆Z.A classB⊆τ is said to be a basis for τ if all nonempty open subset of U can be represented as a union of subfamily of B.Evidently, any topology can have numerous bases, but the basisB generates a unique topology τ.Each union of elements ofB belongs to τ; therefore a basis of τ entirely decides τ.A familyS⊆τ is said to be a subbasis for a topological space U,τ if the collection of all finite intersections of S represents a basis for U,τ.For any classS of subsets of U, S represents a subbasis for a unique basis B which generates a unique topology τ on U such that for each i∈I(2)B=∩iSi:Si∈S,τ=∪iBi:Bi∈B.


### 2.2. Pawlak Rough Set Theory

The current subsection presents some elementary notions pertaining to rough sets given by Pawlak [1].Definition 1.
[1] Consider U is a finite set called universe, and R is an equivalence relation on U; we symbolize U/R to represent the collection of all equivalence classes of R and sR to symbolize an equivalence class in R that contains an element s∈U. Then, the pair AR=U,R is said to be Pawlak’s approximation space and for any L⊆U, we propose the lower and upper approximation of L by R¯L=s∈U:sR⊆L and R¯L=s∈U:sR∩L≠ϕ, respectively. Moreover, L is called a rough set if R¯L≠R¯L. Otherwise, it is an exact set.Definition 2.
[1] Consider AR=U,R is Pawlak approximation space and L⊆U. Therefore, the boundary, positive, and negative regions and the accuracy of approximations of L⊆U are given, respectively, by(3)BNDRL=R¯L−R¯L,POSRL=R¯L,NEGRL=U−R¯L,μRL=R¯LR¯L,whereR¯L≠ϕ.
Properties associated with rough sets can be seen in [1].
It is well known that the set of all definable subsets of the approximation spaceU,R gives rise to a clopen topology τc [8]. In this paper first, we will study how this topology is obtained and why in this topology each open set is closed as well.
AsU/R=sR:s∈U, now, for each A,B∈U/R, A∩B=∅ and U=A∪B. Thus U/R may act as a basis for a τ topology on U.Theorem 1.
If the pairU,R is Pawlak approximation space, then, τc=τ.Proof.
LetA∈τc. Then A is a definable set, so A=A1∪A2∪⋯∪An, where A1,A2,…,An∈U/R. Hence A∈τ. That is τc⊆τ. Conversely, every B∈τ is union of some elements of U/R, which are definable. Since union of definable sets is again definable, B is definable. This means B∈τc. So τ⊆τc as required.Theorem1 explains that topology of definable sets in any Pawlak’s approximation space is produced by the elements of the set U/R. In this topology every open set is closed because complement of any subset in the basis U/R of this topology is the union of all remaining subsets.Study of topology constructed by definable sets helps us to answer some very interesting questions such as the following:What is the probability that a subset ofU may be a definable set?What is the probability that a nonempty subset ofU has an empty lower approximation?What is the probability that a proper nonempty subset ofU has upper approximation equal to U?Answer to the first question is a bit simple and the formula to find the probability that a subset ofU may be a definable set is given as follows:(4)P=τ2n,whereτis the cardinality ofτandnrepresents a number of elements inU.Thus, the probability that a subset of the universeU is a rough set is 1−P.Now, for the answer of the second question first the following result must be considered.Theorem 2.
IfX is a subset of U. Then R¯X is an empty set if and only if X does not contain any nonempty element of τ.Proof.
LetR¯X=∅. Then, by definition, there does not exist any x∈X such that xR⊆X. This implies xR⊈X, for each x. Therefore, for each x∈X,xR⊈X. That is, no element of τ is contained in X. Conversely, let there exist some x∈X, with x∈Ux∈τ such that Ux⊆X. This indicates ∪x∈XUx⊆X. Then by definition R¯X=∪x∈XUx≠∅, which is a contradiction, and therefore, the subset X does not contain any nonempty element of τ.Now in any Pawlak approximation space, the probabilityP∅ that lower approximation of a subset is an empty set can be obtained by the following formula:(5)P∅=1+NXτ2n,whereNXτrepresents a number of subsets ofUwhich does notcontain any nonempty element ofτandnis the number of elements inU.To find the answer to the last question, we may have to consider the following result.Theorem 3.
LetX be a nonempty subset of U. Then R¯X=U if and only if the subset X intersects with every nonempty element of τ.Proof.
LetR¯X=U. Then, by definition, there does not exist any x∈X such that xR∩X=∅. Since nonempty elements of τ are union of some classes xR∈U/R. As x∈∪xR=U, X intersects with every xR. Consequently, it intersects with every nonempty element of τ. Conversely, let there exist some nonempty Ux∈τ containing some x∈U such that Ux∩X=∅. As Ux is the union of some elements of U/R, there exists some class xR∈U/R such that xR∩X=∅. So xR⊈R¯X, which results in R¯X≠U, a contradiction; hence, X intersects with every nonempty element of τ.Further, in any Pawlak approximation space, the probabilityPU that the upper approximation of a subset is U may be obtained by the following formula:(6)PU=NNXτ2n,whereNNXτis the number of subsets ofUwhich intersect with everynonempty element ofτ,andnrepresents the number of elements inU.Example 1.
ConsiderU=1,2,3,4 is a set and R represents an equivalence relation on U, such that U/R=1,2,3,4; then U,R represents Pawlak’s approximation space. Now U/R can be a basis for a topology τ on U. Let us write the topology τ generated by U/R as follows: τ=∅,U,1,2,3,4,1,2,3,1,4,2,3,4. The only definable subsets of U are all elements of τ. Now(7)τ=Number of elements inτ=Number of definable subsets ofU.
Total number of subsets ofU=24=16.
Thus, the probability that a subset ofU is definable is given by(8)τ24=816=0.5.
Next∅, 2, 3 are the only subsets of U which do not contain any nonempty element of τ. Therefore, their lower approximation is empty.(9)P∅=NXτ2n=324=316.
Further the subsets1,2,4, 1,3,4, 1,2,3,4 are the only subsets of U, which intersect with every nonempty element of τ. Therefore their upper approximation is U and then(10)PU=NNXτ2n=324=316.


### 2.3. Soft Set Theory and Soft Rough Sets

Definition 3.
(see [3]). Consider U to be a set of items and E to be a finite set of certain parameters in relative to the objects in U. Parameters represent attributes or characteristics of U objects. A “soft set” on U is the pair F,A, where A⊆E, PU symbolize the power set of U, and F represents the map F:A⟶PA. On the other hand, a soft set over U is a parameterized collection of subsets of U. For e∈A, Fe represents the set of e-approximate elements of a soft set F,A. Note that, sometimes a soft set is indicated by FA and expressed as a set of ordered pairs FA=e,Fe:e∈A.Definition 4.
(see [24]). Consider FA is a soft set on U. Thus, the pair As=U,FA is said to be a soft approximation space. Established on a soft approximation space As, we give the “soft As lower and soft As upper” approximations of X⊆U, respectively, by(11)S¯X=u∈U:∃e∈A,u∈Fe⊆X,S¯X=u∈U:∃e∈A,u∈Fe,Fe∩X≠ϕ.
Generally,S¯X and S¯X refer to soft rough approximations of X⊆U with respect to As. Furthermore, the sets(12)POSAsX=S¯X,NEGAsX=U−S¯X,BNDAsX=S¯X−S¯X,are named the soft “As positive, As negative, and As boundary” regions of X⊆U, individually. Evidently, if S¯X=S¯X=X, i.e., BNDAsX=ϕ, then X⊆U is said to be “soft As definable” or “soft As exact” set; or else X is called a “soft As rough” set. Moreover there may be a subset which has the same lower and upper approximations but is not definable. Besides, we suggest the accuracy of approximations by(13)μAsX=S¯XS¯X,whenS¯X≠ϕ,0,otherwise.
Above definition gives a generalization of Pawlak rough sets theory; naturally it may not satisfy some properties of their properties.Proposition 1.
(see [24]). If FA is a soft set on U and As=U,FA is a soft approximation space, then, for each X⊆U:(14)S¯X=∪e∈AFe:Fe⊆X,S¯X=∪e∈AFe:Fe∩X≠ϕ.
Properties associated with soft rough sets can be gotten in [34].Definition 5.
(see [24]). Suppose that FA is a soft set on U and As=U,FA a soft approximation space. Then, FA is said to be a “full soft set” if FA=∪e∈AFe. It is clear that if FA is a full soft set, then ∀x∈U,∃e∈A such that x∈Fe.Proposition 2.
(see [24]). If FA is a full soft set on U and As=U,FA is a soft approximation space, then, the subsequent conditions are true:(i)
S¯U=S¯U=U(ii)
X⊆S¯X, ∀X⊆U(iii)
S¯x≠ϕ, ∀x∈U
Now, we present and study the idea of the topology of all definable sets generated by soft setFA. Moreover, we illustrate the condition in which this topology is well defined. Now, again, we can find the answers to the questions which are related to the probability associated with subsets of a set in soft rough sets.The following example illustrates that the condition “full soft set” in Proposition2 is necessary to achieve the properties (i)-(iii).Example 2.
ConsiderAs=U,FA to be a soft approximation space, such that U=a,b,c,d, and FA to be a soft set on U where A=e1,e2,e3 and FA=e1,a,e2b,c,e3,a,b. Then, it is clear that FA is not full soft set and thus we have S¯U=S¯U=a,b,c≠U. Also, if X=d, then S¯X=S¯X=ϕ≠X.Theorem 4.
IfFA is a full soft set on U and As=U,FA is a soft approximation space, then, the collection τD=X⊆U:S¯X=S¯X is a quasi-discrete topology on U.Proof.
SinceFA is a soft set on U, by using the properties of the soft approximations in [24], we get(T1) S¯U=S¯U and S¯ϕ=S¯ϕ. Therefore, U,ϕ∈τD.(T2) Let Xi:i∈I be a class of members in τD. Then, ∪i∈IXi∈τD.(T3) Let Xi:i=1,2,3,…,n be a class of finite members in τD. Then, ∩i=1nXi∈τD. Now, we need to prove that τD is a quasi-discrete as follows: Let X∈τD; then S¯X=S¯X. By taking the complement to both sides, we obtain S¯Xc=S¯Xc and this implies S¯Xc=S¯Xc. Thus, Xc∈τD.Remark 1.
According to Proposition2, the condition “full soft set” in the above theorem is necessary to construct the topology τD. If the soft set is not full then U may not be a definable set. Moreover there may be a subset which has same lower and upper approximations but is not definable.
For any soft approximation spaceAs=U,FA, such that FA be a full soft set, the probability of a subset of U is proposed by
P=τD/2n, where τD is the cardinality of τD and n represents a number of elements in U.
Thus, the probability that a subset ofU is a rough set is 1−P.
Now, for the answer of the second question, first the following results must be considered.Lemma 1.
Suppose thatFA is a full soft set on U. Hence, we get(1)
∀x∈U, ∃Bx∈τD such that x∈Bx and Bx is called an open set containing x.(2)
∀x∈U, ∃e∈A such that x∈Fe.(3)
∀e∈A, ∃Bx∈τD such that Fe⊆Bx.Proof.
The proof of (1) is obvious and from the definition ofS¯ and S¯ (Definition 4), the proofs of (2) and (3) are straightforward.
Now, for any soft approximation spaceAs=U,FA, such that FA is a full soft set, the probability P∅ that lower approximation of a subset is an empty set can be obtained by the following formula:(15)P∅=1+Λ∅FA2n,whereΛ∅FAis the number of subsets ofUwhich does notcontain any nonempty element ofFAandnrepresents a number of elements inU.
In order to find the answer of the last question we may have to consider the following.Theorem 5.
ConsiderAs=U,FA is a soft approximation space such that FA is a full soft set and X is a subset of U. Thus, S¯X=U if and only if X intersects with every nonempty element of τD.Proof.
Firstly, sinceFA is a full soft set, then, ∀e∈A, ∪e∈AFe=U. Let S¯X=U; then S¯X=∪e∈AFe such that Fe∩X≠ϕ. Therefore, by Lemma 4, ∃Bx∈τD such that Fe⊆Bx and Fe∩X≠ϕ. Accordingly, ∀Bx∈τD, Bx∩X≠ϕ which means that X intersects with every nonempty element of τD. Conversely, let ∃Bx∈τD such that Bx∩X=ϕ and S¯X≠U. Thus ∃e∈A such that Fe⊆Bx and Fe∩X=ϕ. Accordingly, S¯X≠U which is a contradiction.
Note that ifS¯X=U, then no need for X to intersect with Fe,∀e∈A in general as Example 3 illustrates. Clearly, the subsets b,c and b,c,d do not intersect with Fe1=a although their soft upper approximation is U.
Further in any soft approximation space,As=U,FA, such that FA is a full soft set; the probability PU that upper approximation of a subset is U can be obtained by the following formula:(16)PU=ΛXτD2n,whereΛXτDis the number of subsets ofUwhich intersect withevery nonempty element ofτandnis the number of elements inU.
The next example explains the previous discussion.Example 3.
ConsiderAs=U,FA is a soft approximation space, such that U=a,b,c,d and FA is a full soft set on U, where A=e1,e2,e3,e4 and FA=e1,a,e2,a,b,e3,c,e4,b,d. Then, we obtain the topology of all soft definable subsets by τD=U,ϕ,c,a,b,d. Therefore, τD=number of elements in τD= number of all definable subsets of U. Thus, the probability that a subset of U is definable is given by P=τD/2n=τD/2n=4/24=4/16=1/4=0.25.
Now, the subsetsb and d are the only subsets of U that do not contain any nonempty element of FA. Therefore, their lower approximation is empty.
Accordingly,P∅=1+Λ∅FA/2n=1+2/24=3/16. Further, the subsets b,c,a,b,c,a,c,d, and b,c,d are the only subsets of U, which intersect with every nonempty element of τD. Therefore, their upper approximation is U and accordingly PU=ΛXτD/2n=4/24=4/16=1/4=0.25.


## 2.1. Topological Space

A topology [4] of a set U is defined by the collection τ of subsets of U which fulfills the following three axioms:(T1) ϕ, U∈τ.(T2) A finite intersection of subsets of τ is a member in τ.(T3) An arbitrary union of subsets of τ is a member in τ.We call a pairU,τ “topological space” or “space” and the members of U “points” of τ, and the subsets of U that belong to τ are said to be “open” sets and the complements of the open sets are called “closed” sets in the space. The collection of all closed sets denotes τc.An interiorintAresp.,closureclA of a subset A is given by a union of all open sets contained in Aresp., intersection of all closed sets that containA, formally:(1)intA=∪V∈τ:V⊆A,clA=∩Z∈τc:A⊆Z.A classB⊆τ is said to be a basis for τ if all nonempty open subset of U can be represented as a union of subfamily of B.Evidently, any topology can have numerous bases, but the basisB generates a unique topology τ.Each union of elements ofB belongs to τ; therefore a basis of τ entirely decides τ.A familyS⊆τ is said to be a subbasis for a topological space U,τ if the collection of all finite intersections of S represents a basis for U,τ.For any classS of subsets of U, S represents a subbasis for a unique basis B which generates a unique topology τ on U such that for each i∈I(2)B=∩iSi:Si∈S,τ=∪iBi:Bi∈B.


## 2.2. Pawlak Rough Set Theory

The current subsection presents some elementary notions pertaining to rough sets given by Pawlak [1].Definition 1.
[1] Consider U is a finite set called universe, and R is an equivalence relation on U; we symbolize U/R to represent the collection of all equivalence classes of R and sR to symbolize an equivalence class in R that contains an element s∈U. Then, the pair AR=U,R is said to be Pawlak’s approximation space and for any L⊆U, we propose the lower and upper approximation of L by R¯L=s∈U:sR⊆L and R¯L=s∈U:sR∩L≠ϕ, respectively. Moreover, L is called a rough set if R¯L≠R¯L. Otherwise, it is an exact set.Definition 2.
[1] Consider AR=U,R is Pawlak approximation space and L⊆U. Therefore, the boundary, positive, and negative regions and the accuracy of approximations of L⊆U are given, respectively, by(3)BNDRL=R¯L−R¯L,POSRL=R¯L,NEGRL=U−R¯L,μRL=R¯LR¯L,whereR¯L≠ϕ.
Properties associated with rough sets can be seen in [1].
It is well known that the set of all definable subsets of the approximation spaceU,R gives rise to a clopen topology τc [8]. In this paper first, we will study how this topology is obtained and why in this topology each open set is closed as well.
AsU/R=sR:s∈U, now, for each A,B∈U/R, A∩B=∅ and U=A∪B. Thus U/R may act as a basis for a τ topology on U.Theorem 1.
If the pairU,R is Pawlak approximation space, then, τc=τ.Proof.
LetA∈τc. Then A is a definable set, so A=A1∪A2∪⋯∪An, where A1,A2,…,An∈U/R. Hence A∈τ. That is τc⊆τ. Conversely, every B∈τ is union of some elements of U/R, which are definable. Since union of definable sets is again definable, B is definable. This means B∈τc. So τ⊆τc as required.Theorem1 explains that topology of definable sets in any Pawlak’s approximation space is produced by the elements of the set U/R. In this topology every open set is closed because complement of any subset in the basis U/R of this topology is the union of all remaining subsets.Study of topology constructed by definable sets helps us to answer some very interesting questions such as the following:What is the probability that a subset ofU may be a definable set?What is the probability that a nonempty subset ofU has an empty lower approximation?What is the probability that a proper nonempty subset ofU has upper approximation equal to U?Answer to the first question is a bit simple and the formula to find the probability that a subset ofU may be a definable set is given as follows:(4)P=τ2n,whereτis the cardinality ofτandnrepresents a number of elements inU.Thus, the probability that a subset of the universeU is a rough set is 1−P.Now, for the answer of the second question first the following result must be considered.Theorem 2.
IfX is a subset of U. Then R¯X is an empty set if and only if X does not contain any nonempty element of τ.Proof.
LetR¯X=∅. Then, by definition, there does not exist any x∈X such that xR⊆X. This implies xR⊈X, for each x. Therefore, for each x∈X,xR⊈X. That is, no element of τ is contained in X. Conversely, let there exist some x∈X, with x∈Ux∈τ such that Ux⊆X. This indicates ∪x∈XUx⊆X. Then by definition R¯X=∪x∈XUx≠∅, which is a contradiction, and therefore, the subset X does not contain any nonempty element of τ.Now in any Pawlak approximation space, the probabilityP∅ that lower approximation of a subset is an empty set can be obtained by the following formula:(5)P∅=1+NXτ2n,whereNXτrepresents a number of subsets ofUwhich does notcontain any nonempty element ofτandnis the number of elements inU.To find the answer to the last question, we may have to consider the following result.Theorem 3.
LetX be a nonempty subset of U. Then R¯X=U if and only if the subset X intersects with every nonempty element of τ.Proof.
LetR¯X=U. Then, by definition, there does not exist any x∈X such that xR∩X=∅. Since nonempty elements of τ are union of some classes xR∈U/R. As x∈∪xR=U, X intersects with every xR. Consequently, it intersects with every nonempty element of τ. Conversely, let there exist some nonempty Ux∈τ containing some x∈U such that Ux∩X=∅. As Ux is the union of some elements of U/R, there exists some class xR∈U/R such that xR∩X=∅. So xR⊈R¯X, which results in R¯X≠U, a contradiction; hence, X intersects with every nonempty element of τ.Further, in any Pawlak approximation space, the probabilityPU that the upper approximation of a subset is U may be obtained by the following formula:(6)PU=NNXτ2n,whereNNXτis the number of subsets ofUwhich intersect with everynonempty element ofτ,andnrepresents the number of elements inU.Example 1.
ConsiderU=1,2,3,4 is a set and R represents an equivalence relation on U, such that U/R=1,2,3,4; then U,R represents Pawlak’s approximation space. Now U/R can be a basis for a topology τ on U. Let us write the topology τ generated by U/R as follows: τ=∅,U,1,2,3,4,1,2,3,1,4,2,3,4. The only definable subsets of U are all elements of τ. Now(7)τ=Number of elements inτ=Number of definable subsets ofU.
Total number of subsets ofU=24=16.
Thus, the probability that a subset ofU is definable is given by(8)τ24=816=0.5.
Next∅, 2, 3 are the only subsets of U which do not contain any nonempty element of τ. Therefore, their lower approximation is empty.(9)P∅=NXτ2n=324=316.
Further the subsets1,2,4, 1,3,4, 1,2,3,4 are the only subsets of U, which intersect with every nonempty element of τ. Therefore their upper approximation is U and then(10)PU=NNXτ2n=324=316.


## 2.3. Soft Set Theory and Soft Rough Sets

Definition 3.
(see [3]). Consider U to be a set of items and E to be a finite set of certain parameters in relative to the objects in U. Parameters represent attributes or characteristics of U objects. A “soft set” on U is the pair F,A, where A⊆E, PU symbolize the power set of U, and F represents the map F:A⟶PA. On the other hand, a soft set over U is a parameterized collection of subsets of U. For e∈A, Fe represents the set of e-approximate elements of a soft set F,A. Note that, sometimes a soft set is indicated by FA and expressed as a set of ordered pairs FA=e,Fe:e∈A.Definition 4.
(see [24]). Consider FA is a soft set on U. Thus, the pair As=U,FA is said to be a soft approximation space. Established on a soft approximation space As, we give the “soft As lower and soft As upper” approximations of X⊆U, respectively, by(11)S¯X=u∈U:∃e∈A,u∈Fe⊆X,S¯X=u∈U:∃e∈A,u∈Fe,Fe∩X≠ϕ.
Generally,S¯X and S¯X refer to soft rough approximations of X⊆U with respect to As. Furthermore, the sets(12)POSAsX=S¯X,NEGAsX=U−S¯X,BNDAsX=S¯X−S¯X,are named the soft “As positive, As negative, and As boundary” regions of X⊆U, individually. Evidently, if S¯X=S¯X=X, i.e., BNDAsX=ϕ, then X⊆U is said to be “soft As definable” or “soft As exact” set; or else X is called a “soft As rough” set. Moreover there may be a subset which has the same lower and upper approximations but is not definable. Besides, we suggest the accuracy of approximations by(13)μAsX=S¯XS¯X,whenS¯X≠ϕ,0,otherwise.
Above definition gives a generalization of Pawlak rough sets theory; naturally it may not satisfy some properties of their properties.Proposition 1.
(see [24]). If FA is a soft set on U and As=U,FA is a soft approximation space, then, for each X⊆U:(14)S¯X=∪e∈AFe:Fe⊆X,S¯X=∪e∈AFe:Fe∩X≠ϕ.
Properties associated with soft rough sets can be gotten in [34].Definition 5.
(see [24]). Suppose that FA is a soft set on U and As=U,FA a soft approximation space. Then, FA is said to be a “full soft set” if FA=∪e∈AFe. It is clear that if FA is a full soft set, then ∀x∈U,∃e∈A such that x∈Fe.Proposition 2.
(see [24]). If FA is a full soft set on U and As=U,FA is a soft approximation space, then, the subsequent conditions are true:(i)
S¯U=S¯U=U(ii)
X⊆S¯X, ∀X⊆U(iii)
S¯x≠ϕ, ∀x∈U
Now, we present and study the idea of the topology of all definable sets generated by soft setFA. Moreover, we illustrate the condition in which this topology is well defined. Now, again, we can find the answers to the questions which are related to the probability associated with subsets of a set in soft rough sets.The following example illustrates that the condition “full soft set” in Proposition2 is necessary to achieve the properties (i)-(iii).Example 2.
ConsiderAs=U,FA to be a soft approximation space, such that U=a,b,c,d, and FA to be a soft set on U where A=e1,e2,e3 and FA=e1,a,e2b,c,e3,a,b. Then, it is clear that FA is not full soft set and thus we have S¯U=S¯U=a,b,c≠U. Also, if X=d, then S¯X=S¯X=ϕ≠X.Theorem 4.
IfFA is a full soft set on U and As=U,FA is a soft approximation space, then, the collection τD=X⊆U:S¯X=S¯X is a quasi-discrete topology on U.Proof.
SinceFA is a soft set on U, by using the properties of the soft approximations in [24], we get(T1) S¯U=S¯U and S¯ϕ=S¯ϕ. Therefore, U,ϕ∈τD.(T2) Let Xi:i∈I be a class of members in τD. Then, ∪i∈IXi∈τD.(T3) Let Xi:i=1,2,3,…,n be a class of finite members in τD. Then, ∩i=1nXi∈τD. Now, we need to prove that τD is a quasi-discrete as follows: Let X∈τD; then S¯X=S¯X. By taking the complement to both sides, we obtain S¯Xc=S¯Xc and this implies S¯Xc=S¯Xc. Thus, Xc∈τD.Remark 1.
According to Proposition2, the condition “full soft set” in the above theorem is necessary to construct the topology τD. If the soft set is not full then U may not be a definable set. Moreover there may be a subset which has same lower and upper approximations but is not definable.
For any soft approximation spaceAs=U,FA, such that FA be a full soft set, the probability of a subset of U is proposed by
P=τD/2n, where τD is the cardinality of τD and n represents a number of elements in U.
Thus, the probability that a subset ofU is a rough set is 1−P.
Now, for the answer of the second question, first the following results must be considered.Lemma 1.
Suppose thatFA is a full soft set on U. Hence, we get(1)
∀x∈U, ∃Bx∈τD such that x∈Bx and Bx is called an open set containing x.(2)
∀x∈U, ∃e∈A such that x∈Fe.(3)
∀e∈A, ∃Bx∈τD such that Fe⊆Bx.Proof.
The proof of (1) is obvious and from the definition ofS¯ and S¯ (Definition 4), the proofs of (2) and (3) are straightforward.
Now, for any soft approximation spaceAs=U,FA, such that FA is a full soft set, the probability P∅ that lower approximation of a subset is an empty set can be obtained by the following formula:(15)P∅=1+Λ∅FA2n,whereΛ∅FAis the number of subsets ofUwhich does notcontain any nonempty element ofFAandnrepresents a number of elements inU.
In order to find the answer of the last question we may have to consider the following.Theorem 5.
ConsiderAs=U,FA is a soft approximation space such that FA is a full soft set and X is a subset of U. Thus, S¯X=U if and only if X intersects with every nonempty element of τD.Proof.
Firstly, sinceFA is a full soft set, then, ∀e∈A, ∪e∈AFe=U. Let S¯X=U; then S¯X=∪e∈AFe such that Fe∩X≠ϕ. Therefore, by Lemma 4, ∃Bx∈τD such that Fe⊆Bx and Fe∩X≠ϕ. Accordingly, ∀Bx∈τD, Bx∩X≠ϕ which means that X intersects with every nonempty element of τD. Conversely, let ∃Bx∈τD such that Bx∩X=ϕ and S¯X≠U. Thus ∃e∈A such that Fe⊆Bx and Fe∩X=ϕ. Accordingly, S¯X≠U which is a contradiction.
Note that ifS¯X=U, then no need for X to intersect with Fe,∀e∈A in general as Example 3 illustrates. Clearly, the subsets b,c and b,c,d do not intersect with Fe1=a although their soft upper approximation is U.
Further in any soft approximation space,As=U,FA, such that FA is a full soft set; the probability PU that upper approximation of a subset is U can be obtained by the following formula:(16)PU=ΛXτD2n,whereΛXτDis the number of subsets ofUwhich intersect withevery nonempty element ofτandnis the number of elements inU.
The next example explains the previous discussion.Example 3.
ConsiderAs=U,FA is a soft approximation space, such that U=a,b,c,d and FA is a full soft set on U, where A=e1,e2,e3,e4 and FA=e1,a,e2,a,b,e3,c,e4,b,d. Then, we obtain the topology of all soft definable subsets by τD=U,ϕ,c,a,b,d. Therefore, τD=number of elements in τD= number of all definable subsets of U. Thus, the probability that a subset of U is definable is given by P=τD/2n=τD/2n=4/24=4/16=1/4=0.25.
Now, the subsetsb and d are the only subsets of U that do not contain any nonempty element of FA. Therefore, their lower approximation is empty.
Accordingly,P∅=1+Λ∅FA/2n=1+2/24=3/16. Further, the subsets b,c,a,b,c,a,c,d, and b,c,d are the only subsets of U, which intersect with every nonempty element of τD. Therefore, their upper approximation is U and accordingly PU=ΛXτD/2n=4/24=4/16=1/4=0.25.


## 3. Topological Soft Rough Approximations of Soft Rough Sets

The current section is devoted to introduction of topological soft rough approximations in view of topological structure. Firstly, it will be seen that soft sets and topological spaces have a very close relationship. The concept of topological soft rough approximations will be presented and their properties will be studied. On the other hand, it will be shown that accuracy of the proposed approach is better than existing techniques. Besides, we will give answers to some important questions about the probability in topological soft rough sets.Definition 6.
ConsiderFA is a soft set on U and K=∪Fe. Thus, we propose the following:(i)
SFA=Fe:e∈A may denote a subbasis;(ii)
BFA=A∩B:A,B∈SFA×SFA may denote a basis for the topology TSR defined as the following.
IfFA is a soft set on U and K=∪Fe∈SFAFe, then the topology TSR can be defined on K with a basis BFA. That is, TSR=∪B:B∈BFA. This topology may be called topology generated by FA and we call it “soft rough topology” (in brief, SR topology).Remark 2.
There are three cases of a subbasisSFA:(i)
IfSFA is a partition of U, then SFA=BFA and will be a basis for a quasi-discrete (clopen) topology (in which all open sets are closed).(ii)
IfSFA is a covering (not partition) of U, then SFA≠BFA and BFA will be a basis for a general topology.(iii)
IfSFA is not a covering (not partition) of U, then SFA≠BFA and BFA will be a basis for a general topology. The following examples explain Remark 2.Example 4.
IfAs=U,FA is a soft approximation space, such that U=a,b,c,d and FA is a soft set on U where A=e1,e2,e3 and FA=e1,a,b,e2,c,e3,d, then, we get that the subbasis of TSR is SFA=c,d,a,b and the basis is BFA=c,d,a,b. Clearly, SFA=BFA. Accordingly, the topology generated by BFA is(17)TSR=U,ϕ,c,d,a,b,c,d,a,b,c,a,b,d.
Obviously,TSR is a quasi-discrete topology.Example 5.
Suppose thatAs=U,FA is a soft approximation space, such that U=a,b,c,d and FA is a soft set on U where A=e1,e2,e3 and FA=e1,a,b,e2,a,c,e3,d. Then, the subbasis of TSR is SFA=d,a,b,a,c and the basis is BFA=a,d,a,b,a,c. Evidently, SFA≠BFA. Accordingly, the topology TSR generated by BFA is(18)TSR=U,ϕ,a,d,a,b,a,c,a,d,a,b,c,a,b,d,a,c,d.
Obviously,TSR is not a quasi-discrete topology.Example 6.
ConsiderAs=U,FA is a soft approximation space, such that U=a,b,c,d and FA is a soft set on U where A=e1,e2,e3 and FA=e1,a,e2,b,c,e3,a,c. Then, the subbasis of TSR is SFA=a,a,c,b,c and the basis is BFA=a,c,a,c,b,c. Undoubtedly, SFA≠BFA. Accordingly, the topology TSR generated by BFA is(19)TSR=U,ϕ,a,c,a,c,b,c,a,b,c.
Obviously,TSR is not a quasi-discrete topology.Definition 7.
ConsiderAs=U,FA is a soft approximation space and TSR is the SR topology on U. The triple ATSR=U,FA,TSR is said to be a “topological soft rough approximation space” (briefly, TSR approximation space).Definition 8.
ConsiderATSR=U,FA,TSR to be a TSR approximation space. Therefore, for each X⊆U we suggest the topological soft rough approximations, “TSR lower” and “TSR upper,” respectively, by(20)S¯TSRX=∪G∈TSR:G⊆X,S¯TSRX=∩H∈TSRc:X⊆H.Remark 3.
(i)
In general,S¯TSRX and S¯TSRX represent the interior and closure of X associated with the topology TSR, respectively.(ii)
IfSFA is a partition on U, then SFA=BFA=U/R and hence S¯TSRX and S¯TSRX are identical with Pawlak’s rough set approximations. Therefore, it can be said that the proposed approach is equivalent to Pawlak’s approach only in case SFA is a partition of U. Accordingly, we can say that Pawlak’s rough set model is a specialization of proposed model. Example 4 illustrated this fact.(iii)
IfSFA is not a partition on U, then SFA≠BFA≠U/R and hence S¯TSRX and S¯TSRX will be different from Pawlak’s approximations as illustrated in Example 7.Example 7.
Suppose thatU=a,b,c,d is a set of students reading some languages. Let A=e1,e2,e3 and As=U,FA be a soft approximation space, where FA is a soft set on U. Consider the next information system in Table 1.
Therefore, the equivalence classes areU/R=b,d,a,c.
If we consider each attribute of the setEnglish,French,German represents a parameter as the following: e1=English, e2=French and e3=German, then, we get the following:
The soft set ofU is FA=e1,a,c,d,e2,b,d,e3,a,c, where A=e1,e2,e3. Thus, we get SFA=a,c,b,d,a,c,d and the basis is BFA=d,a,c,b,d,a,c,d Accordingly, the topology TSR generated by BFA is(21)TSR=U,ϕ,d,a,c,b,d,a,c,d,and the complement of TSR is(22)TSRc=U,ϕ,b,a,c,b,d,a,b,c.
Evidently, theTSR approximations of any subsets of U differ than Pawlak’s rough approximations.Table 1
The tabular form for a soft setF,A.
EnglishFrenchGermanaYesNoYesbNoYesNocYesNoYesdYesYesNoDefinition 9.
Suppose thatATSR=U,FA,TSR is a TSR approximation space. Hence, for every X⊆U, we express the “TSR positive, TSR negative, and TSR boundary” regions and the “TSR accuracy” of the TSR approximations, respectively, by(23)POSTSRX=S¯TSRX,NEGTSRX=U−S¯TSRX,BNDTSRX=S¯TSRX−S¯TSRX,μTSRX=S¯TSRXS¯TSRX,ifX≠ϕ,1,ifXis aTSR−definable.Remark 4.
(i)
It is clear that0≤μTSRX≤1, for any X⊆U.(ii)
IfS¯TSRX=S¯TSRX, then BNDTSRX=ϕ and μTSRX=1. Thus X⊆U is said to be “TSR definable” or “TSR exact” set; otherwise X is called a “TSR rough” set.
The core objective of the following propositions is to discuss the basic properties ofTSR rough approximations S¯TSR and S¯TSR.
According to the characteristics of the interior and closure, we can demonstrate the subsequent results, so we omit the proof.Proposition 3.
IfATSR=U,FA,TSR is a TSR approximation space and X,Y⊆U, then, the TSR lower and TSR upper approximations operators satisfy the next properties:(i)
S¯TSR∅=S¯TSR∅=∅.(ii)
S¯TSRU=S¯TSRU=U.(iii)
S¯TSRX⊆X⊆S¯TSRX.(iv)
IfX⊆Y, then S¯TSRX⊆S¯TSRY.(v)
IfX⊆Y, then S¯TSRX⊆S¯TSRY.(vi)
S¯TSRXc=S¯TSRXc.(vii)
S¯TSRX∩Y=S¯TSRX∩S¯TSRY.(viii)
S¯TSRX∪Y=S¯TSRX∪S¯TSRY.(ix)
S¯TSRX∩Y⊆S¯TSRX∩S¯TSRY.(x)
S¯TSRX∪Y⊇S¯TSRX∪S¯TSRY.(xi)
IfX∈TSR, then S¯TSRX=X.(xii)
IfX∈TSRc, then S¯TSRX=X.Proposition 4.
IfATSR=U,FA,TSR is a TSR approximation space and X⊆U, then:(i)
S¯TSRS¯TSRX=S¯TSRX.(ii)
S¯TSRS¯TSRX=S¯TSRX.(iii)
S¯TSRX⊆S¯TSRS¯TSRX.(iv)
S¯TSRS¯TSRX⊆S¯TSRX.Remark 5.
The inclusion relations in Proposition4 may be strict, as shown in Example 8.Example 8.
Suppose thatFA is a soft set on U and As=U,FA is a soft approximation space, where U=a,b,c,d and A=e1,e2,e3,e4,e5 such that
FA=e1,b,e2,a,b,e3,c,d,e4,a,c,d,e5,a,b,c. Therefore, we get the following.
The subbasis ofTSR is SFA=b,a,b,c,d,a,c,d,a,b,c.
The basis ofTSR is BFA=a,b,c,a,b,a,c,c,d,a,c,d,a,b,c.
Accordingly, we get(24)TSR=U,ϕ,a,b,c,a,b,b,c,a,c,c,d,b,c,d,a,c,d,a,b,c.
The complement ofTSR is(25)TSRc=U,ϕ,a,b,d,a,b,a,d,b,d,c,d,b,c,d,a,c,d,a,b,d.
Thus,ATSR=U,S,TSR is a TSR approximation space. Now, let X=a,c and Y=a,b,d. Then, S¯TSRX=a,c and then S¯TSRS¯TSRX=a,c,d , which means that S¯TSRX≠S¯TSRS¯TSRX. Also, S¯TSRY=a,b,d and S¯TSRS¯TSRY=a,b, which means that S¯TSRY≠S¯TSRS¯TSRY.
The following theorem establishes a relationship between approximations of a set in soft rough sets [24] and topological soft rough sets.Theorem 6.
IfATSR=U,FA,TSR is a TSR approximation space and X⊆U, then:(i)
S¯X⊆S¯TSRX.(ii)
S¯TSRX⊆S¯X.Proof:.
We shall verify only the first item and the other likewise. Letx∈S¯X; then ∃e∈A, such that x∈Fe⊆X and Fe∈SFA. Accordingly, Fe∈TSR such that x∈Fe and Fe⊆X; this implies Fe⊆S¯TSRX, and therefore, x∈S¯TSRX. Hence, S¯X⊆S¯TSRX.Corollary 1.
IfATSR=U,FA,TSR is a TSR approximation space and X⊆U, then:(i)
BNDTSRX⊆BNDAsX(ii)
μAsX≤μTSRXCorollary 2.
IfATSR=U,FA,TSR is a TSR approximation space and X⊆U and if X is a soft exact set, then it is a TSR exact set.Remark 6.
(1)
According to the above results, it is easy to see that boundary region in case of topological soft rough sets is smaller than boundary for soft rough sets. Therefore accuracy has been improved in proposed technique.(2)
In general reverse inclusions do not hold in case of Theorem6. Following example explains this.Example 9.
According to Example8: Let X=b,c,d. Thus S¯X=b,c,d and S¯X=U; therefore BNDAsX=a and μAsX=3/4. But, S¯TSRX=S¯TSRX=X and thus BNDTSRX=ϕ and μTSRX=1. Obviously, X is a TSR exact (definable) set (according to our approach) although it is a soft rough set.
Next we define some very important notions.Definition 10.
ConsiderATSR=U,FA,TSR is a TSR approximation space and X⊆U. Thus, we describe the subsequent four elementary sorts of TSR soft rough sets as follows:
The subsetX represents(i)
a roughlyTSR definable set if S¯TSRX≠ϕ and S¯TSRX≠U,(ii)
an internallyTSR indefinable set if S¯TSRX=ϕ and S¯TSRX≠U,(iii)
an externallyTSR indefinable set if S¯TSRX≠ϕ and S¯TSRX=U,(iv)
a totallyTSR indefinable set if S¯TSRX=ϕ and S¯TSRX=U.
The axiomatic significance of this classification is given as follows:(i)
If the subsetX is a roughly TSR definable set, then we can identify for some members of U that they belong to X, and for other members of U that they belong to Xc, by using existing knowledge from the TSR approximation space ATSR.(ii)
If the subsetX is an internally TSR indefinable set, then we can identify about some members of U that they belong to Xc, but we cannot identify for any member of U that it belongs to X, by using ATSR.(iii)
If the subsetX is an externally TSR indefinable set, then we can identify for some members of U that they belong to X, but we cannot identify for any member of U that it belongs to Xc, by using ATSR.(iv)
If the subsetX is a totally TSR indefinable set, then we cannot identify for any member of U whether it belongs to X or Xc, by using ATSR.Theorem 7.
ConsiderATSR=U,FA,TSR to be a TSR approximation space and X⊆U. Therefore, we have the following.(i)
If the subsetX is a roughly TSR definable set, then X is roughly soft As definable.(ii)
If the subsetX is an internally TSR definable set, then X is internally soft As indefinable.(iii)
If the subsetX is an externally TSR definable set, then X is externally soft As indefinable.(iv)
If the subsetX is a totally TSR indefinable set, then X is totally soft As indefinable.Proof.
Only, the first statement will be proved and the other statements can be made by a similar way. (i) Suppose that the subsetX is a roughly soft As definable set; then S¯TSRX≠ϕ and S¯TSRX≠U. Therefore, by using Theorem 6, S¯X≠ϕ and S¯X≠U and thus X represents a roughly soft As definable set.Remark 7.
(i)
Theorem7 shows that soft rough approximations of a set given in [24] are different from TSR rough approximations proposed in this paper. Moreover, it clarifies the significance of the proposed approach in defining approximations of sets; for example, let X be a totally soft As indefinable set. Then, we get S¯X=ϕ and S¯X=U. Thus, we are incapable of identifying what are the elements of U that belong to X or Xc. But, by using TSR rough approximations, it may be S¯TSRX≠ϕ and S¯TSRX≠U which means that X can be roughly TSR definable set and accordingly we can determine for some elements of U that they belong to X, and meanwhile, for some elements of U, we can identify that they belong to Xc, by using the existing information from the TSR approximation space (to illustrate this, see Examples 8 and Subsection 4.1).(ii)
The inverse of Theorem7 does not hold, generally, as demonstrated in Example 9 and Subsection 4.1.
Now, once again, we can find the probability for different types of subsets in topological soft rough sets. Firstly, to answer the first question, we consider the following results.Theorem 8.
IfFA is a soft set on U, ATSR=U,FA,TSR is a TSR approximation space, and X⊆U, then, the class τD=X⊆U:S¯TSRX=S¯TSRX gives rise a topology on U.Proof.
According to Proposition3, the proof is clear.
Note thatτD represents a topology of all definable sets in U.Lemma 2.
IfATSR=U,FA,TSR is a TSR approximation space, then, τD⊆TSR.Proof:.
LetX∈τD; then S¯TSRX=S¯TSRX and this implies S¯TSRX=X. Accordingly, X∈TSR.Lemma 3.
LetATSR=U,FA,TSR be a TSR approximation space and SFA be a partition of U; then τD=TSR and SFA is a basis for τD.Proof.
Firstly, using Lemma2, τD⊆TSR. Now, let X∈TSR which is a quasi-discrete topology; then X is an open and closed subset. Therefore, S¯TSRX=S¯TSRX=X and this implies X∈τD. Hence, TSR⊆τD.
The subsequent example explains that the condition “SFA is a partition of U” is necessary condition.Example 10.
Consider Example7; we get τD=U,ϕ,a,c,b,d but TSR=U,ϕ,d,a,c,b,d,a,c,d. It is clear that TSR⊈τD.Definition 11.
Suppose thatATSR=U,FA,TSR is a TSR approximation space and τD is a topology of all definable sets in U. The probability PD that a subset of U is definable is defined by(26)PD=τD2n,where τD is the cardinality of τD and n represents a number of elements of U.
Therefore, the probability that a subset is a rough setX⊆U is 1−PD.Example 11.
(1)
Consider Example4; we have TSR=τD. Thus, the probability PD that a subset of the universe set is definable is given by(27)PD=τD2n=824=12=0.5.(2)
Consider Example5; we have τD=U,ϕ,d,a,b,c. Therefore, the probability PD that a subset of the universe set is definable is given by(28)PD=τD2n=424=14=0.25.
Secondly, to identify the probability that lower approximation of a nonempty subset ofU may be an empty set, we propose the next results.Lemma 4.
IfATSR=U,FA,TSR is a TSR approximation space, then, BFA⊆TSR.Proof.
The proof is straightforward.Theorem 9.
ConsiderATSR=U,FA,TSR to be a ATSR=U,FA,TSR approximation space and X⊆U. S¯TSRX=ϕ if and only if the subset X does not contain any nonempty element of TSR.Proof.
Using Definition8, the proof is clear.Theorem 10.
LetATSR=U,FA,TSR be a ATSR=U,FA,TSR approximation space and X⊆U. S¯TSRX=ϕ if and only if the proper subsets of X are not element in BFA.Proof.
Firstly, ifS¯TSRX=ϕ then ∀G⊆X, G∉TSR, and this implies ∀G⊆X, G∉BFA. Conversely, let ∃G⊆X, G∈BFA such that S¯TSRX=ϕ. Then, G∈TSR such that G⊆X and this implies G⊆S¯TSRX which contradicts assumption S¯TSRX=ϕ. Accordingly, X does not contain any nonempty element of TSR.Definition 12.
Suppose thatATSR=U,FA,TSR is a TSR approximation space. The probability P∅ that the TSR lower approximation of a subset of the universe set is an empty set is defined by(29)P∅=1+Λ∅TSR2n,whereΛ∅TSRrepresents a number of subsets ofUwhich does notcontain any nonempty element ofTSRandnis the number of elements inU.
Therefore, the probability that theTSR lower approximation of X⊆U is not an empty set is 1−P∅.Example 12.
(1)
Consider Example4; obviously, the subsets a and b are the only subsets which do not contain any nonempty element of TSR. Thus, the probability P∅ that the TSR lower approximation of a subset of the universe set is an empty set is defined by(30)P∅=1+Λ∅TSR2n=1+224=316=0.19.(2)
Consider Example5; obviously, the subsets b,c, and b,c are the only subsets which do not contain any nonempty element of TSR. Therefore, the probability Pϕ that the TSR lower approximation of a subset of the universe set is an empty set is defined by(31)Pϕ=1+Λ∅TSR2n=1+324=416=0.25.
In order to find the probability thatTSR upper approximation of any subset is U, we may have to suggest the subsequent result.Theorem 11.
ConsiderATSR=U,FA,TSR is a TSR approximation space and X⊆U. S¯X=U if and only if the subset X intersects with every element in TSR.Proof.
Obvious.
Now, for anyTSR approximation space, the probability PU that TSR upper approximation of a subset is U can be obtained by the following formula:(32)PU=ΛXTSR2n,whereΛXTSRis the number of subsets ofUwhich intersect withevery nonempty element ofTSRandnis the number of elements inU.
The next example explains the above discussion.Example 13.
(1)
Consider Example4; obviously, the subsets a,c,d and b,c,d are the only subsets which intersect with every nonempty element of TSR. Thus, the probability PU that TSR upper approximation of a subset is U is defined by(33)PU=ΛXTSR2n=224=216=0.125.(2)
Consider Example5; obviously, the subsets a,d,a,b,d, and a,c,d are the only subsets which intersect with every nonempty element of TSR. Thus, the probability PU that TSR upper approximation of a subset is U is defined by(34)PU=ΛXTSR2n=324=316=0.19.


## 4. Medical Application in Heart Failure

In the current article, we illustrate the significance of the suggested approach in decision-making problems for medical applications. Consequently, we apply it to the issue of heart failure. We have a data set with the results of five symptoms for twenty patients divided into twelve malesp3,p6,p8,p9,p11,…,p17,p19 and 8 females p1,p2,p4,p5,p7,p10,p18,p20. The study was conducted at Om El-Kora Cardiac Center, Hospital of Heart Diseases, Tanta, Egypt. This research involved twenty patients who came to the hospital with various symptoms and underwent a thorough history, physical examination, lab tests, resting ECG, and conventional echo assessment. Finally, the diagnosis of heart failure was verified.


### 4.1. The Experimental Results

The experimental findings are discussed in this subsection by adding a preparatory analysis performed on five heart disease symptoms for twenty patients, according to Thivagar and Richard [40]. Table 2 shows the data from the information system for twenty patients, addressing the heart failure issue. The columns reflect the signs of heart failure diagnosis (where “Yes” indicates that the patient has symptoms and “No” indicates that the patient has none) [40], with condition attributes, such that e1 indicates “the breathlessness,” e2 indicates “the orthopnea,” e3 indicates “the paroxysmal nocturnal dyspnea,” e4 indicates “reduced exercise tolerance,” and e5 indicates “the ankle swelling.” Attribute D indicates “decision of heart failure.” For rows in Table 2, P=p1,p2,p3,…,p20 represents the set of twenty patients. Therefore, the set of all attributes is A=e1,e2,e3,e4,e5∪D which is represented by columns.Table 2
Original medical information system.
U/Ae1e2e3e4e5Dp111000Yesp200011Nop301101Yesp411110Yesp500010Nop600011Nop710010Nop810010Nop901101Yesp1000000Nop1110011Nop1201101Nop1311111Yesp1410100Yesp1510011Nop1600011Nop1701100Yesp1811101Yesp1901101Yesp2011011YesHere 1 and 0 denote “yes” and “no”, respectively.We apply the suggested method in the set of female’s patients only and the others similarly.Accordingly, Table3 represents the soft set of female’s patients, where the set of female’s patients is U=p1,p2,p4,p5,p7,p10,p18,p20 and the set of attributes is A=e1,e2,e3,e4,e5.Table 3
The tabular form for female’s soft setF,A.
U/Ae1e2e3e4e5Dp111000Yesp200011Nop411110Yesp500010Nop710010Nop1000000Nop1811101Yesp2001101YesLetF,A be a soft set over given by Table 3; the basis, generated by F,A, is given by(35)BFA=p4,p18,p2,p20,p4,p18,p4,p20,p18,p20,p7,p18,p20,p2,p4,p5,p20,p2,p7,p18,p20,p1,p4,p18,p20.Therefore, the topology generated by this base is(36)TSR=U,ϕ,p2,p4,p18,p2,p4,p2,p18,p4,p7,p4,p18,p18,p20,p1,p4,p18,p2,p4,p7,p2,p4,p18,p2,p18,p20,p4,p7,p18,p4,p18,p20,p1,p2,p4,p18,p1,p4,p7,p18,p1,p4,p18,p20,p2,p4,p5,p7,p2,p4,p18,p20,p4,p7,p18,p20,p1,p2,p4,p7,p18,p1,p2,p4,p18,p20,p1,p4,p7,p18,p20,p2,p4,p5,p7,p18,p2,p4,p7,p18,p20,p1,p2,p4,p5,p7,p18,p1,p2,p4,p7,p18,p20,p2,p4,p5,p7,p18,p20,p1,p2,p4,p5,p7,p18,p20.The complement ofTSR is(37)TSRc=U,ϕ,p10,p1,p10,p5,p10,p10,p20,p1,p5,p10,p1,p10,p20,p2,p5,p10,p5,p7,p10,p5,p10,p20,p1,p5,p10,p1,p5,p7,p10,p1,p10,p18,p20,p2,p5,p7,p10,p2,p5,p10,p20,p5,p7,p10,p20,p1,p2,p5,p7,p10,p1,p2,p5,p10,p20,p1,p5,p7,p10,p20,p1,p4,p5,p7,p10,p1,p5,p10,p18,p20,p2,p5,p7,p10,p20,p1,p2,p4,p5,p7,p10,p1,p2,p5,p10,p18,p20,p1,p2,p5,p7,p10,p20,p1,p4,p5,p7,p10,p20,p1,p5,p7,p10,p18,p20,p1,p2,p4,p5,p7,p10,p20,p1,p2,p5,p7,p10,p18,p20,p1,p4,p5,p7,p10,p18,p20.Now, we introduce a comparison between the boundary and the accuracy of the approximations using the proposed approximations “TSR approximations” and the previous ones in Table 4.(1)
Table4 presents a comparison between the soft rough sets [24] and topological soft rough sets.(2)
For a comparison with the method given in [26], we have the following classes for each parameter:Fore1 classes are p1,p4,p7,p18, p2,p5,p10,p20Fore2 classes are p1,p4,p18,p20, p2,p5,p7,p10Fore3 classes are p4,p18,p20, p1,p2,p5,p7,p10Fore4 classes are p2,p4,p5,p7, p1,p10,p18,p20Fore5 classes are p2,p18,p20, p1,,p4,p5,p7,p10Now, the intersection of all these classes isp1, p2, p4, p5, p7, p10, p18, p20. This means intersection of all equivalence relations gives rise to identity relation R=p1,p1,p2,p2,p4,p4,p5,p5,p7,p7,p10,p10,p18,p18,p20,p20. Therefore, every subset will be definable in approximation space U,R. Therefore, the method discussed in [26] fails to produce distinct lower and upper approximations in the given example.(3)
For a comparison with the method given in [25], we have the map φ:U⟶PE as the following φp1=e1,e2, φp2=e4,e5, φp4=e1,e2,e3,e4, φp5=e4, φp7=e1,e4, φp10=, φp18=e1,e2,e3,e5, φp20=e2,e3,e5. Clearly these sets are distinct. Therefore, we define the approximation of any subset X of U as per the following:(38)X¯φ=x∈X,φx≠φyfor ally∈Xc,X¯φx∈X,φx=φyfor somey∈X.Table 4
Comparisons among some soft rough approximations andTSR approximations.
XSoft rough set approach [24]The suggested approachBNDASXμASXBNDTSRXμTSRXp4p1,p2,p4,p5,p7,p18,p200p1,p5,p101/4p10ϕ0p100p4,p10U−p100p41/5p2,p4,p10U−p100p1,p5,p7,p101/2p2,p10,p18,p20p1,p4,p5,p7,p183/7p1,p5,p101/2p2,p4,p5,p7,p10,p20p1,p18,p204/7p1,p10,p204/7So, all subsets ofU will be definable so this method also fails to produce distinct lower and upper approximations in the given example.Decision-making is essential in the daily lives, and this process yields the best alternative from a variety of options. We give Algorithm1 in table for a decision-making of an information system in terms of the TSR approximations.Algorithm 1: A decision-making via TSR approximations.
Step1: Input the soft set F,A.Step2: Take the class SFA=Fe:∀e∈A as a subbasis for a basis BFA.Step3: Compute the basis BFA=A∩B:A,B∈SFA×SFA by Definition 6.Step4: Generate the topology TSR=∪B:B∈BFA by Definition 6.Step5: Investigate the TSR upper approximations, say S¯TSRX, and TSR lower approximations, say S¯TSRX, for every X⊆U, according to Definition 8.Step6: Determine the boundary region, say BNDTSRX, from Step 2, according to Definition 9.Step7: Calculate the accuracy of the approximation, say μTSRX, from Step 2, according to Definition 9.Step8: Decide, exactly, rough sets and exact sets, using Definition 9.


### 4.2. Reduction of Attributes

A very important purpose of rough sets is the reduction of data by removing redundant attributes in the information system. So, the present subsection is devoted for the reduction of an information system in case of topological rough sets for the data given in Table3. We extend the notion of “nanotopology,” which has proposed by Thivagar and Richard [40], to TSR approximations. We shall apply the nanotopology of TSR approximations to identify the key factors of “heart failure” using topological reduction of attributes in information system of Table 3.First, let us extend the definition of “nanotopology” into “TSR nanotopology” using “TSR approximations.”Definition 13.
ConsiderATSR=U,FA,TSR is a TSR approximation space, and X⊆U. Therefore, the class NTSR=U,ϕ,S¯TSRX,S¯TSRX,BNDTSRX is called “TSR nanotopology” which represents a general topology generated by the soft rough set X⊆U. The basis of this topology is given the class βTSR=U,S¯TSRX,BNDTSRX.Definition 14.
ConsiderATSR=U,FA,TSR is a TSR approximation space, and NTSR is a TSR nanotopology with a basis βTSR. Then,(i)
ifβTSR−ek=βTSR, then the attribute ek is called “dispensable”;(ii)
ifβTSR−ek≠βTSR, then the attribute ek is not “dispensable.” Therefore, the core of attributes is CORE=ek which represents the common part of reduction.
Now, we apply the topological reduction for Table3 to identify the key factors of “heart failure” as follows: We compute the TSR nanotopology to decision-making for two sets of patients:
X=p1,p4,p18,p20 which represents a set of patients that have the disease of heart failure and Y=p2,p5,p7,p10 which represents a set of patients that do not have the disease of heart failure.
We will make a topological reduction for first setX and the second set Y similarly.Case 1.
(patients having the heart failure disease).
According to Table3, we get S¯TSRX=p1,p4,p18,p20, S¯TSRX=U−p2, and BNDTSRX=p5,p7,p10. Thus, the basis of TSR nanotopology generated by the above TSR approximations is(39)βTSR=U,p1,p4,p18,p20,p5,p7,p10.Step 1.
When the attribute “the breathlessnesse1” is removed:
The topology generated by this base is(40)TSR=U,ϕ,p2,p4,p20,p2,p4,p2,p20,p4,p20,p18,p20,p2,p18,p20,p4,p18,p20,p2,p4,p5,p7,p1,p4,p18,p20,p2,p4,p18,p20,p1,p2,p4,p18,p20,p2,p4,p5,p7,p20,p2,p4,p5,p7,p18,p20,p1,p2,p4,p7,p18,p20.
The complement ofTSR is(41)TSRc=U,ϕ,p10,p1,p10,p1,p10,p18,p5,p7,p10,p1,p5,p7,p10,p2,p5,p7,p10,p1,p10,p18,p20,p1,p2,p5,p7,p10,p1,p4,p5,p7,p10,p1,p2,p4,p5,p7,p10,p1,p2,p5,p7,p10,p18,p1,p4,p5,p7,p10,p18,p1,p5,p7,p10,p18,p20,p1,p2,p4,p5,p7,p10,p18,p1,p2,p5,p7,p10,p18,p20,p1,p4,p5,p7,p10,p18,p20.
Therefore,TSR approximations of X in this case are
S¯TSRX=p1,p4,p18,p20, S¯TSRX=U−p2, and BNDTSRX=p5,p7,p10. Thus, the basis of TSR nanotopology generated by the above TSR approximations is(42)βTSR−e1=U,p1,p4,p18,p20,p5,p7,p10=βTSR.Step 2.
When the attribute “the orthopneae2” is removed:
By the same way as in Step1, we get(43)βTSR−e2=U,p4,p18,p1,p5,p7,p10,p20≠βTSR.Step 3.
When the attribute “the paroxysmal nocturnal dyspneae3” is removed: By the same way as in Step 1, we get(44)βTSR−e3=U,p1,p4,p18,p20,p5,p7,p10=βTSR.Step 4.
When the attribute “reduced exercise tolerancee4” is removed:
By the same way as in Step1, we get(45)βTSR−e4=U,p1,p4,p18,p20,p2,p5,p7,p10≠βTSR.Step 5.
When the attribute “the ankle swellinge5” is removed:
By the same way as in Step1, we get(46)βTSR−e5=U,p1,p4,p18,p20,p2,p5,p7,p10≠βTSR.
Therefore, we get the attributese1,e3 are dispensable and e2,e4,e5 are not dispensable. Accordingly, the core of attributes is CORETSR=e2,e4,e5, i.e., “the orthopnea, reduced exercise tolerance, and the ankle swelling” represent the main attributes that have close joining to the disease of the heart failure.
By the same manner, we can make a topological reduction to Table2 (an information system of all “the heart failure” patients).
At the end of the paper, we give Algorithm2 in table which can be used to make a topological reduction of attributes for information system in terms of the TSR approximations.
Algorithm 2: Topological reduction of attributes via TSR approximations.
Step1: Input a soft set F,A, by using the two finite sets, a universe U and A a set of attributes (parameters). which represent the data as an information table, rows of which are labeled by attributes A and columns by objects, and entries of the table are attribute values.Step2: Compute the TSR upper approximations, say S¯TSRX, and TSR lower approximations, say S¯TSRX, and TSR boundary, say BNDTSRX, for the decision set X⊆U, according to Definition 8.Step3: Generate the base βTSR of TSR nanotopology generated by TSR approximations in Step 2 using Definition 13.Step4: Eliminate an attribute ek from the attributes A and find the TSR upper approximation, TSR lower approximation, and TSR boundary for the decision set X on A−ek.Step5: Generate the base βTSR−ek of TSR nanotopology generated by TSR approximations given in Step 4 using Definition 13.Step6: Recurrence Steps 4 and 5 for every attribute in A.Step7: The attributes in A for which βTSR−ek≠βTSR form the CORETSR.
In literature many methods for reduction of parameters for soft sets have been given, for example, [32, 34, 41]. In all these methods only positive parameters are considered which result in a decision parameter as the sum of the values allotted to an alternative. Then on the basis of this decision parameter most suitable alternative is selected. In the present case of decision-making these methods fail. In given case parameters are not all positive but a blend of positive and negative, decided by the experts. Moreover decision parameter is attached already.


## 4.1. The Experimental Results

The experimental findings are discussed in this subsection by adding a preparatory analysis performed on five heart disease symptoms for twenty patients, according to Thivagar and Richard [40]. Table 2 shows the data from the information system for twenty patients, addressing the heart failure issue. The columns reflect the signs of heart failure diagnosis (where “Yes” indicates that the patient has symptoms and “No” indicates that the patient has none) [40], with condition attributes, such that e1 indicates “the breathlessness,” e2 indicates “the orthopnea,” e3 indicates “the paroxysmal nocturnal dyspnea,” e4 indicates “reduced exercise tolerance,” and e5 indicates “the ankle swelling.” Attribute D indicates “decision of heart failure.” For rows in Table 2, P=p1,p2,p3,…,p20 represents the set of twenty patients. Therefore, the set of all attributes is A=e1,e2,e3,e4,e5∪D which is represented by columns.Table 2
Original medical information system.
U/Ae1e2e3e4e5Dp111000Yesp200011Nop301101Yesp411110Yesp500010Nop600011Nop710010Nop810010Nop901101Yesp1000000Nop1110011Nop1201101Nop1311111Yesp1410100Yesp1510011Nop1600011Nop1701100Yesp1811101Yesp1901101Yesp2011011YesHere 1 and 0 denote “yes” and “no”, respectively.We apply the suggested method in the set of female’s patients only and the others similarly.Accordingly, Table3 represents the soft set of female’s patients, where the set of female’s patients is U=p1,p2,p4,p5,p7,p10,p18,p20 and the set of attributes is A=e1,e2,e3,e4,e5.Table 3
The tabular form for female’s soft setF,A.
U/Ae1e2e3e4e5Dp111000Yesp200011Nop411110Yesp500010Nop710010Nop1000000Nop1811101Yesp2001101YesLetF,A be a soft set over given by Table 3; the basis, generated by F,A, is given by(35)BFA=p4,p18,p2,p20,p4,p18,p4,p20,p18,p20,p7,p18,p20,p2,p4,p5,p20,p2,p7,p18,p20,p1,p4,p18,p20.Therefore, the topology generated by this base is(36)TSR=U,ϕ,p2,p4,p18,p2,p4,p2,p18,p4,p7,p4,p18,p18,p20,p1,p4,p18,p2,p4,p7,p2,p4,p18,p2,p18,p20,p4,p7,p18,p4,p18,p20,p1,p2,p4,p18,p1,p4,p7,p18,p1,p4,p18,p20,p2,p4,p5,p7,p2,p4,p18,p20,p4,p7,p18,p20,p1,p2,p4,p7,p18,p1,p2,p4,p18,p20,p1,p4,p7,p18,p20,p2,p4,p5,p7,p18,p2,p4,p7,p18,p20,p1,p2,p4,p5,p7,p18,p1,p2,p4,p7,p18,p20,p2,p4,p5,p7,p18,p20,p1,p2,p4,p5,p7,p18,p20.The complement ofTSR is(37)TSRc=U,ϕ,p10,p1,p10,p5,p10,p10,p20,p1,p5,p10,p1,p10,p20,p2,p5,p10,p5,p7,p10,p5,p10,p20,p1,p5,p10,p1,p5,p7,p10,p1,p10,p18,p20,p2,p5,p7,p10,p2,p5,p10,p20,p5,p7,p10,p20,p1,p2,p5,p7,p10,p1,p2,p5,p10,p20,p1,p5,p7,p10,p20,p1,p4,p5,p7,p10,p1,p5,p10,p18,p20,p2,p5,p7,p10,p20,p1,p2,p4,p5,p7,p10,p1,p2,p5,p10,p18,p20,p1,p2,p5,p7,p10,p20,p1,p4,p5,p7,p10,p20,p1,p5,p7,p10,p18,p20,p1,p2,p4,p5,p7,p10,p20,p1,p2,p5,p7,p10,p18,p20,p1,p4,p5,p7,p10,p18,p20.Now, we introduce a comparison between the boundary and the accuracy of the approximations using the proposed approximations “TSR approximations” and the previous ones in Table 4.(1)
Table4 presents a comparison between the soft rough sets [24] and topological soft rough sets.(2)
For a comparison with the method given in [26], we have the following classes for each parameter:Fore1 classes are p1,p4,p7,p18, p2,p5,p10,p20Fore2 classes are p1,p4,p18,p20, p2,p5,p7,p10Fore3 classes are p4,p18,p20, p1,p2,p5,p7,p10Fore4 classes are p2,p4,p5,p7, p1,p10,p18,p20Fore5 classes are p2,p18,p20, p1,,p4,p5,p7,p10Now, the intersection of all these classes isp1, p2, p4, p5, p7, p10, p18, p20. This means intersection of all equivalence relations gives rise to identity relation R=p1,p1,p2,p2,p4,p4,p5,p5,p7,p7,p10,p10,p18,p18,p20,p20. Therefore, every subset will be definable in approximation space U,R. Therefore, the method discussed in [26] fails to produce distinct lower and upper approximations in the given example.(3)
For a comparison with the method given in [25], we have the map φ:U⟶PE as the following φp1=e1,e2, φp2=e4,e5, φp4=e1,e2,e3,e4, φp5=e4, φp7=e1,e4, φp10=, φp18=e1,e2,e3,e5, φp20=e2,e3,e5. Clearly these sets are distinct. Therefore, we define the approximation of any subset X of U as per the following:(38)X¯φ=x∈X,φx≠φyfor ally∈Xc,X¯φx∈X,φx=φyfor somey∈X.Table 4
Comparisons among some soft rough approximations andTSR approximations.
XSoft rough set approach [24]The suggested approachBNDASXμASXBNDTSRXμTSRXp4p1,p2,p4,p5,p7,p18,p200p1,p5,p101/4p10ϕ0p100p4,p10U−p100p41/5p2,p4,p10U−p100p1,p5,p7,p101/2p2,p10,p18,p20p1,p4,p5,p7,p183/7p1,p5,p101/2p2,p4,p5,p7,p10,p20p1,p18,p204/7p1,p10,p204/7So, all subsets ofU will be definable so this method also fails to produce distinct lower and upper approximations in the given example.Decision-making is essential in the daily lives, and this process yields the best alternative from a variety of options. We give Algorithm1 in table for a decision-making of an information system in terms of the TSR approximations.Algorithm 1: A decision-making via TSR approximations.
Step1: Input the soft set F,A.Step2: Take the class SFA=Fe:∀e∈A as a subbasis for a basis BFA.Step3: Compute the basis BFA=A∩B:A,B∈SFA×SFA by Definition 6.Step4: Generate the topology TSR=∪B:B∈BFA by Definition 6.Step5: Investigate the TSR upper approximations, say S¯TSRX, and TSR lower approximations, say S¯TSRX, for every X⊆U, according to Definition 8.Step6: Determine the boundary region, say BNDTSRX, from Step 2, according to Definition 9.Step7: Calculate the accuracy of the approximation, say μTSRX, from Step 2, according to Definition 9.Step8: Decide, exactly, rough sets and exact sets, using Definition 9.


## 4.2. Reduction of Attributes

A very important purpose of rough sets is the reduction of data by removing redundant attributes in the information system. So, the present subsection is devoted for the reduction of an information system in case of topological rough sets for the data given in Table3. We extend the notion of “nanotopology,” which has proposed by Thivagar and Richard [40], to TSR approximations. We shall apply the nanotopology of TSR approximations to identify the key factors of “heart failure” using topological reduction of attributes in information system of Table 3.First, let us extend the definition of “nanotopology” into “TSR nanotopology” using “TSR approximations.”Definition 13.
ConsiderATSR=U,FA,TSR is a TSR approximation space, and X⊆U. Therefore, the class NTSR=U,ϕ,S¯TSRX,S¯TSRX,BNDTSRX is called “TSR nanotopology” which represents a general topology generated by the soft rough set X⊆U. The basis of this topology is given the class βTSR=U,S¯TSRX,BNDTSRX.Definition 14.
ConsiderATSR=U,FA,TSR is a TSR approximation space, and NTSR is a TSR nanotopology with a basis βTSR. Then,(i)
ifβTSR−ek=βTSR, then the attribute ek is called “dispensable”;(ii)
ifβTSR−ek≠βTSR, then the attribute ek is not “dispensable.” Therefore, the core of attributes is CORE=ek which represents the common part of reduction.
Now, we apply the topological reduction for Table3 to identify the key factors of “heart failure” as follows: We compute the TSR nanotopology to decision-making for two sets of patients:
X=p1,p4,p18,p20 which represents a set of patients that have the disease of heart failure and Y=p2,p5,p7,p10 which represents a set of patients that do not have the disease of heart failure.
We will make a topological reduction for first setX and the second set Y similarly.Case 1.
(patients having the heart failure disease).
According to Table3, we get S¯TSRX=p1,p4,p18,p20, S¯TSRX=U−p2, and BNDTSRX=p5,p7,p10. Thus, the basis of TSR nanotopology generated by the above TSR approximations is(39)βTSR=U,p1,p4,p18,p20,p5,p7,p10.Step 1.
When the attribute “the breathlessnesse1” is removed:
The topology generated by this base is(40)TSR=U,ϕ,p2,p4,p20,p2,p4,p2,p20,p4,p20,p18,p20,p2,p18,p20,p4,p18,p20,p2,p4,p5,p7,p1,p4,p18,p20,p2,p4,p18,p20,p1,p2,p4,p18,p20,p2,p4,p5,p7,p20,p2,p4,p5,p7,p18,p20,p1,p2,p4,p7,p18,p20.
The complement ofTSR is(41)TSRc=U,ϕ,p10,p1,p10,p1,p10,p18,p5,p7,p10,p1,p5,p7,p10,p2,p5,p7,p10,p1,p10,p18,p20,p1,p2,p5,p7,p10,p1,p4,p5,p7,p10,p1,p2,p4,p5,p7,p10,p1,p2,p5,p7,p10,p18,p1,p4,p5,p7,p10,p18,p1,p5,p7,p10,p18,p20,p1,p2,p4,p5,p7,p10,p18,p1,p2,p5,p7,p10,p18,p20,p1,p4,p5,p7,p10,p18,p20.
Therefore,TSR approximations of X in this case are
S¯TSRX=p1,p4,p18,p20, S¯TSRX=U−p2, and BNDTSRX=p5,p7,p10. Thus, the basis of TSR nanotopology generated by the above TSR approximations is(42)βTSR−e1=U,p1,p4,p18,p20,p5,p7,p10=βTSR.Step 2.
When the attribute “the orthopneae2” is removed:
By the same way as in Step1, we get(43)βTSR−e2=U,p4,p18,p1,p5,p7,p10,p20≠βTSR.Step 3.
When the attribute “the paroxysmal nocturnal dyspneae3” is removed: By the same way as in Step 1, we get(44)βTSR−e3=U,p1,p4,p18,p20,p5,p7,p10=βTSR.Step 4.
When the attribute “reduced exercise tolerancee4” is removed:
By the same way as in Step1, we get(45)βTSR−e4=U,p1,p4,p18,p20,p2,p5,p7,p10≠βTSR.Step 5.
When the attribute “the ankle swellinge5” is removed:
By the same way as in Step1, we get(46)βTSR−e5=U,p1,p4,p18,p20,p2,p5,p7,p10≠βTSR.
Therefore, we get the attributese1,e3 are dispensable and e2,e4,e5 are not dispensable. Accordingly, the core of attributes is CORETSR=e2,e4,e5, i.e., “the orthopnea, reduced exercise tolerance, and the ankle swelling” represent the main attributes that have close joining to the disease of the heart failure.
By the same manner, we can make a topological reduction to Table2 (an information system of all “the heart failure” patients).
At the end of the paper, we give Algorithm2 in table which can be used to make a topological reduction of attributes for information system in terms of the TSR approximations.
Algorithm 2: Topological reduction of attributes via TSR approximations.
Step1: Input a soft set F,A, by using the two finite sets, a universe U and A a set of attributes (parameters). which represent the data as an information table, rows of which are labeled by attributes A and columns by objects, and entries of the table are attribute values.Step2: Compute the TSR upper approximations, say S¯TSRX, and TSR lower approximations, say S¯TSRX, and TSR boundary, say BNDTSRX, for the decision set X⊆U, according to Definition 8.Step3: Generate the base βTSR of TSR nanotopology generated by TSR approximations in Step 2 using Definition 13.Step4: Eliminate an attribute ek from the attributes A and find the TSR upper approximation, TSR lower approximation, and TSR boundary for the decision set X on A−ek.Step5: Generate the base βTSR−ek of TSR nanotopology generated by TSR approximations given in Step 4 using Definition 13.Step6: Recurrence Steps 4 and 5 for every attribute in A.Step7: The attributes in A for which βTSR−ek≠βTSR form the CORETSR.
In literature many methods for reduction of parameters for soft sets have been given, for example, [32, 34, 41]. In all these methods only positive parameters are considered which result in a decision parameter as the sum of the values allotted to an alternative. Then on the basis of this decision parameter most suitable alternative is selected. In the present case of decision-making these methods fail. In given case parameters are not all positive but a blend of positive and negative, decided by the experts. Moreover decision parameter is attached already.


## 5. Conclusion

In this article notion of topological soft rough sets is introduced, where topology generated from a soft set plays a vital role. Here notion of soft rough approximations is discussed and some of their properties are given. Their properties have been studied and their relationships with some other methods have been examined. In fact, the proposed approaches fulfill all axioms of Pawlak’s rough sets without adding extra restrictions as Propositions3 and 4 illustrated. The proposed techniques depend basically on general topology and hence they open the way for applications of topology in soft rough sets. Further, we have answered some very important questions, such as how to determine the probability that a subset of the universe is definable in the classical rough sets and their extensions (like the soft sets and topological soft rough sets).Finally, we have introduced medical applications, in the decision-making of medical diagnosis for heart failure problems [39], to illustrate the importance of current methods and also to compare proposed method and the previous ones. Moreover, we have succeeded in making a topological reduction for the data set covering the result of five symptoms for twenty patients with heart failure disease, and thus we identify the core factors of the heart failure diagnosis. Besides, two algorithms to our method have been obtained.For future works, it is hoped that presented framework may be useful to study its application in COVID-19 and other diseases.



---

*Source: 2559495-2021-12-07.xml*