A topological reduction for predicting of a lung cancer disease based on generalized rough sets

Abstract

The present work proposes new styles of rough sets by using different neighborhoods which are made from a general binary relation. The proposed approximations represent a generalization to Pawlak’s rough sets and some of its generalizations, where the accuracy of these approximations is enhanced significantly. Comparisons are obtained between the methods proposed and the previous ones. Moreover, we extend the notion of “nano-topology”, which have introduced by Thivagar and Richard [49], to any binary relation. Besides, to demonstrate the importance of the suggested approaches for deciding on an effective tool for diagnosing lung cancer diseases, we include a medical application of lung cancer disease to identify the most risk factors for this disease and help the doctor in decision-making. Finally, two algorithms are given for decision-making problems. These algorithms are tested on hypothetical data for comparison with already existing methods.

Keywords

Neighborhoods topology rough sets generalized nano-topology attributes reduction and lung cancer disease

1 Introduction

Rough set theory is a modern, non-statistical approach to deal with uncertainty and vagueness proposed by Pawlak [1, 2]. This theory presents a logical and comprehensible view, to deal with vagueness and uncertainty in the data collected from real-life situations. There are many applications of rough sets ranging from algebra to decision-making problems [3 –13]. The essential thinking of this theory is built simply on the indiscernibility and discernibility of objects. The indiscernibility relation, which represents a relation of equivalence, induces a space of approximation made up, of classes of equivalence of indiscernible objects. The core of the Pawlak approach is the approximations of rough sets by using the equivalence relation on the domain. However, the restrictions on the equivalence relations in Pawlak rough sets cause some problems and limitations of theoretical and practical aspects. So, many proposals have been made for generalizing these assumptions [14 –33].

In several fields of science and engineering, for instance, Chemistry, Biology, Image processing, Information acquirement, and Pattern recognition, Topology, and Rough set theories were applied. As a result, how to conglomerate rough set philosophy and topology structure becomes an important and likely research subject that has significant attention from specialists in this community [34 –40]. In particular, this topic was explored separately by Skowron [41] and Wiweger [42] in 1988. This subject continued to be discussed by Lin and developed a linking amongst the fuzzy rough sets and topology [43]. Besides, based on the philosophies of topology and neighborhood systems, Polkowski [44] established and categorized topological spaces based on information systems. In [45], Polkowski mentions that “aspects of the rough set theory have been recognized at an early stage within the framework of the topology of partitions”. Zhu [23] has studied some cover-based rough sets via a topological vision. In 2019, Zhan et al. [34, 35] presented different types of soft coverings-based rough sets and applied them in decision-making problems. Yang [46] examined the relationship between the separation axioms and the two topological spaces. Kortelainen [47] and Jarvinen [48] explored the relationship between modified sets, topological spaces, and rough pre-order sets. The notion of interior (resp. closure) operator in topology represents a counterpart of lower (resp. upper) approximation in rough sets [17]. Some other authors discussed the relationship between the topology and generalized rough sets from different perspectives (for instance, see [17, 33]). The main contribution of the present research is to combine some philosophies in terms of the conceptions of general topology (namely, nano-topology [49]) and applied them in rough set theory. In fact, the notion of nano-topology represents one of the topological applications in rough set theory. So, a generalized nano-topology is proposed for any generalized rough sets to extend the application fields of nano-topology. Additionally, we illustrate the conditions that any generalized rough sets generate a nano-topology. Moreover, we explain how to generate generalized nano-topology from any arbitrary binary relation.

The notion of adhesion-set was first introduced in covering-based rough sets [50]. After then, Nawar et al. [51] used it to define new generalized neighborhoods (so-called j-adhesion neighborhoods) generated from any binary relation and thus they suggested new different sorts of covering-based rough sets. Besides, based on the concept of j-neighborhood space [22], M. Atef et al. [52] introduced different six rough approximations generated by j-adhesion neighborhoods. In the present paper, we use this concept to define a new generalized neighborhood, generating via a binary relation. Therefore, new generalized approximations, called “adhesion-approximations”, of rough sets are constructed. We introduced more properties of adhesion-neighborhood and results different essentially than [51] and [52]. Besides, the suggested approximations are based on a general binary relation and establish a partition for the objects without any extra conditions on the relation. Therefore, all properties of Pawlak’s rough sets are held in general which was never realized in any other method. Accordingly, this technique extends the application fields of rough set theory in any real-life problem.

The remainder of the paper is prepared as follows. Some of Pawlak’s basic concepts and some of its generalizations are briefly reviewed in Section 2. Besides, we present an improvement for Allam et al.’s approaches [19] as a modification for their approximations. We will prove that the condition of reflexivity of the relation needn’t be held to satisfy Pawlak’s properties in these methods.

In Section 3, we extend the concept of “adhesion set” to a new generalized neighborhood induced by a binary relation. Based on this neighborhood, new generalized rough sets are proposed and their properties are studied. Comparisons are also made between the proposed method and the previous methods. Besides, we will confirm that the previous methods (Yao [18] and Allam et al. [19]) failed in satisfying the basic properties of Pawlak’s rough sets in the general case of the binary relation. Therefore, these approaches restrict the application fields of the theory of rough set. Additionally, we demonstrate that the proposed method is stronger than Yao’s approach in the case of reflexive relation. Based on this relation, we illustrate that the suggested method and Allam’s approach are independent that is the two approaches are not comparable in general.

Section 4 is devoted to generalizing the notion of “nano-topology” introduced by Thivagar and Richard, to any binary relation. An important result to illustrate the conditions that generate generalized nano-topology from arbitrary generalized rough sets is proposed. Furthermore, in Section 5, we use generalized nano-topology and apply the proposed methods to medical applications (in fact, we use real-life data collected from an experiment involving forty-five patients who undergo treatment at the Nanjing Chest Hospital Respiratory Department, China [53]) generated through binary relations. In this application, we use a general binary relation to illustrate the significance of the suggested technique in decision-making. Therefore, Pawlak, Yao, and Allam’s methods can’t apply here and hence we can say that our method extends the application field of rough sets. Comparisons between the proposed approaches and the previous approaches are examined. Besides, we are introducing a topological reduction, using the proposed methods, to identify the most risk factors for lung cancer disease. Additionally, two algorithms are provided that can be used for decision-making problems in the information system. Finally, Section 6 concludes with a few comments.

2 Preliminaries

We present some of the basic concepts and results of Pawlak’s rough sets, as well as some of its generalizations, in this section.

2.1 Pawlak’s rough sets

Definition 2.1 [54] Consider the binary relation R on the non-empty set U. Then, an after set (resp. fore set) of an element x ∈ U is defined by a class xR ={ y ∈ U : xRy } (resp. Rx ={ y ∈ U : yRx }).

Definition 2.2 [54] The binary relation R on a set U is said to be

Serial, if for every x ∈ U, ∃y ∈ U such that xRy, i. e. xR≠ ∅.

Inverse serial, if for every x ∈ U, ∃y ∈ U such that yRx, i. e. x ∈ yR.

Reflexive, if for every x ∈ U, xRx.

Symmetric: If for every x, y ∈ U and xRy, then yRx.

Transitive, if for every x, y, z ∈ U, xRy and yRz, then xRz.

Pre-order, if it is a reflexive and transitive relation.

Equivalence, if it is a reflexive, symmetric and transitive relation.

Definition 2.3 [1] Consider R be an equivalence relation on a finite set U (called a universe), and let U/R be the family of all equivalence classes of R such that [x] _R denotes to an equivalence class in R that contains an element x ∈ U. Then, the pair (U, R) is called Pawlak approximation space. Pawlak defined, for any X ⊆ U, the lower and upper approximations of X as follows: $\underline{Apr} (X) = {x \in X : {[x]}_{R} \subseteq X}$ and $\bar{Apr} (X) = {x \in U : {[x]}_{R} \cap X \neq \emptyset}$ , respectively.

Note that: According to Pawlak’s definition, X is called an exact set if $\underline{Apr} (X) = \bar{Apr} (X)$ . Otherwise, X is called a rough set.

Definition 2.4 [1] Suppose that (U, R) be Pawlak approximation space and X ⊆ U. Pawlak proposed the boundary, positive and negative regions and the accuracy of the approximations of X ⊆ U, respectively, as follows: $\begin{matrix} {BND}_{R} (X) = \bar{Apr} (X) - \underline{Apr} (X), \\ {POS}_{R} (X) = \underline{Apr} (X), \\ {NEG}_{R} (X) = U - \bar{Apr} (X), and \\ μ_{R} (X) = \frac{| \underline{Apr} (X) |}{| \bar{Apr} (X) |}, where \bar{Apr} (X) \neq \emptyset . \end{matrix}$

Proposition 2.1 [1] Consider ∅ to be the empty set and let X^c represents a complement of X in U. The followings are held:

$\underline{Apr} (X) \subseteq X$ .

$\underline{Apr} (\emptyset) = \emptyset$ .

$\underline{Apr} (U) = U$ .

If X ⊆ Y, then $\underline{Apr} (X) \subseteq \underline{Apr} (Y)$ .

$\underline{Apr} (X \cap Y) = \underline{Apr} (X) \cap \underline{Apr} (Y)$ .

$\underline{Apr} (X) \cup \underline{Apr} (Y) \subseteq \underline{Apr} (X \cup Y)$ .

$\underline{Apr} (X^{c}) = (\bar{R} (X))^{c}$ .

$\underline{Apr} (\underline{Apr} (X)) = \underline{Apr} (X)$ .

$\underline{Apr} (\bar{Apr} (X)) = \bar{Apr} (X)$ .

If K ∈ U/R, then $\underline{Apr} (K) = K$ .

$X \subseteq \bar{Apr} (X)$ .

$\bar{Apr} (\emptyset) = \emptyset$ .

$\bar{Apr} (U) = U$ .

If X ⊆ Y, then $\bar{Apr} (X) \subseteq \bar{Apr} (Y)$ .

$\bar{Apr} (X \cup Y) = \bar{Apr} (X) \cup \bar{Apr} (Y)$ .

$\bar{Apr} (X) \cap \bar{Apr} (Y) \supseteq \bar{Apr} (X \cap Y)$ .

$\bar{Apr} (X^{c}) = (\underline{Apr} (X))^{c}$ .

$\bar{Apr} (\bar{Apr} (X)) = \bar{Apr} (X)$ .

$\bar{Apr} (\underline{Apr} (X)) = \underline{Apr} (X)$ .

If K ∈ U/R, then $\bar{Apr} (K) = K$ .

2.2 Yao’s rough sets

Yao has proposed lower and upper approximations using neighborhoods that have resulted from a general binary relation, as illustrated in the following definitions.

Definition 2.5 [18] Consider R to be a binary relation on the universe U. We say that y is related to x by R, if xRy and then y represents a successor of x and thus x is a predecessor of y. For any x ∈ U, we define its successor neighborhood by: xR ={ y ∈ U : xRy }.

Definition 2.6 [18] Suppose that R represents any binary relation on the universe U. For every X ⊆ U, we give its lower and upper approximations respectively by: $\begin{matrix} \underline{R} (X) = {x \in U : xR \subseteq X} and \\ \bar{R} (X) = {x \in U : xR \cap X \neq \emptyset} . \end{matrix}$

Besides, the boundary region of X and the accuracy of the approximations is given, respectively, by $B_{R} (X) = \bar{R} (X) - \underline{R} (X) and δ_{R} (X) = \frac{| \underline{R} (X) |}{| \bar{R} (X) |} .$

According to [18], the above approximations satisfied the properties (L3-L7), and (U2, U4-U7) in a general case. The remainders of the properties of Pawlak are achieved only in some special cases of relations as demonstrated in the following result.

Theorem 2.1 [18] Let R be a pre-order relation on U and X ⊆ U. Then, the lower and upper approximations of X, defined by Definition 2.5, satisfy all properties of Pawlak’s rough sets in Proposition 2.1.

2.3 Allam et al.’s rough sets

Definition 2.7 [19] Consider R to be a binary relation on the universe U. For each x ∈ U, we give its minimal neighborhood as follows: $〈 x 〉 R = {\begin{matrix} \underset{x \in yR}{\cap} yR if \exists y such that x \in yR, \\ \emptyset Otherwise . \end{matrix}$

Definition 2.8 [19] Consider R to be a binary relation on the universe U. For every X ⊆ U, the lower and upper approximations are given, respectively, by $\begin{matrix} L (X) = {x \in U : 〈 x 〉 R \subseteq X} and \\ U (X) = {x \in U : 〈 x 〉 R \cap X \neq \emptyset} . \end{matrix}$

In addition, the boundary region of X and the accuracy of the approximations is given, respectively, by $B_{〈 R 〉} (X) = U (X) - L (X) and δ_{〈 R 〉} (X) = \frac{| L (X) |}{| U (X) |} .$

According to [19], the above approximations satisfied the properties (L3-L7), and (U2, U4-U7) in a general case. The remainders of Pawlak’s properties are achieved only in some special cases of relations as the next result explains.

Theorem 2.2 [19] Let R be a reflexive relation on the universe U and X ⊆ U. Then, the lower and upper approximations of X, defined by Definition 2.8, satisfied the properties of Pawlak’s rough sets in Proposition 2.1.

Remark 2.1 The properties (L1), (L2), (U1), and (U3) do not true, in the general case, as the following example illustrates.

Example 2.1 Let U ={ a, b, c, d } and R = {(a, a), (a, b) , (b, b) , (b, c) , (c, b) , (c, c)} be a binary relation on U. Then aR ={ a, b }, bR = cR ={ b, c } and dR =∅. Hence, we obtain: aR ={ a, b }, bR ={ b }, cR ={ b, c } and dR =∅. Now, if A ={ a, b, c } and B ={ a, c, d }, then $L (A) = U ⊊ A$ , and $U (B) = {a, c} ⊉ B$ . Besides, we notice that $L (\emptyset) = {d} \neq \emptyset$ and $U (U) = {a, b, c} \neq U$ .

In the following theorem, we generalize and improve Allam et al.’s approach to any inverse serial relation. In fact, we demonstrate that the condition of reflexivity of the relation is not necessary to achieve the properties of Pawlak

Theorem 2.3 If R is an inverse serial relation on U, then, for each X ⊆ U, the followings are held:

(L1) $L (X) \subseteq X$	(U1) $X \subseteq U (X)$
(L2) $L (\emptyset) = \emptyset$	(U3) $U (U) = U$
(L8) $L (L (X)) = L (X)$	(U8) $U (U (X)) = U (X)$

Proof. Only, the first statement will be proved and the other can be made similarly.

Firstly, if R is an inverse serial relation on U. Then, x ∈ xR, for each x ∈ U ... (1)

(L1) Let $x \in L (X)$ , then xR ⊆ X. Thus, by (1), x ∈ X and hence, $L (X) \subseteq X$ . ■

It is easy to prove the following lemma using Definition 2.2, so the proof is omitted.

Lemma 2.1 If R is a reflexive relation on U, then it is a serial and an inverse serial relation.

Note that: Any reflexive relation is an inverse serial; the converse is not true in general as shown in the following example.

Example 2.2 Let U ={ a, b, c, d } and R = {(a, a), (b, c) , (b, d) , (c, a) , (c, b) , (d, a)} be an inverse relation on U. Obviously, R is not reflexive. Moreover, we have:

aR ={ a }, bR ={ a, b } and cR = dR ={ c, d }.

Hence, we notice the following:

For each x ∈ U, x ∈ xR.

For each X ⊆ U, $L (X) \subseteq X \subseteq U (X)$ .

$L (U) = U (U) = U$ and $L (\emptyset) = U (\emptyset) = \emptyset$ .

3 Generalized rough sets based on adhesion neighborhoods via a binary relation

In this section, using the notion of (adhesion-set) defined in the covering-based rough sets [50], a new generalized neighborhood is defined and their properties are studied. Based on this neighborhood, new generalized rough sets are proposed as a generalization of Pawlak rough sets. We will prove that the suggested approach represents the natural extension of Pawlak’s rough set approaches via a binary relation, and therefore all properties of Pawlak’s approach are held within the proposed approach without adding any additional restrictions.

Definition 3.1 [51, 52] Consider R to be any binary relation on U. The adhesion-neighborhood of x ∈ U is given by the set h (x) ={ y ∈ U : yR = xR } and the family of all adhesion-neighborhoods for each element in U is given by the class: $H (U) = {h (x) : x \in U} .$

Note that: For each x ∈ U, x ∈ h (x).

Lemma 3.1 Let R be any binary relation on U. Then:

y ∈ h (x) if and only if h (x) = h (y).

The class $H (U) = {h (x) : x \in U}$ is a partition of U.

Proof: We will prove (i), and the other similarly.

(i) Firstly, if y ∈ h (x), then yR = xR... (1)

Now, let $z \in h (x) \Leftrightarrow zR = xR \overset{(1)}{\Leftrightarrow} zR = yR \Leftrightarrow z \in h (y)$ . Hence, h (x) = h (y). ■

In the following discussion, we suggest new generalized rough sets based on the adhesion neighborhoods, which induced from a binary relation, as a generalization to Pawlak’s rough set models, Yao’s method, and some of the other methods. We propose rough set approximations and prove that these approximations satisfy all the properties of Pawlak’s rough sets without any restrictions on the relation.

Definition 3.2 Consider R to be any binary relation on U. For every X ⊆ U, we propose the adhesion-lower and adhesion-upper approximation, respectively, as follows: $\begin{matrix} \underline{H} (X) = {x \in U : h (x) \subseteq X} and \\ \bar{H} (X) = {x \in U : h (x) \cap X \neq \emptyset} . \end{matrix} .$

Definition 3.3 Consider R to be any binary relation on U. For every X ⊆ U, we propose the adhesion-positive, negative and boundary, region and the accuracy of adhesion approximations, respectively, as follows: $\begin{matrix} ℘ os (X) = \underline{H} (X), \\ N eg (X) = U - \bar{H} (X), \\ B nd (X) = \bar{H} (X) - \underline{H} (X) and \\ λ (X) = \frac{| \underline{H} (X) |}{| \bar{H} (X) |}, where \bar{H} (X) \neq \emptyset . \end{matrix},$

Lemma 3.2 Consider R to be any binary relation on U. For every X ⊆ U, the adhesion-lower and adhesion-upper approximation can be defined respectively by: $\begin{matrix} \underline{H} (X) = \cup {h (x) : x \in U, h (x) \subseteq X}, and \\ \bar{H} (X) = \cup {h (x) : x \in U, h (x) \cap X \neq \emptyset} . \end{matrix}$

Proof. By using Lemma 3.1, the proof is obvious. ■

The following results introduce the basic properties of adhesion-approximations.

Theorem 3.1 Let R be a binary relation on U, and X, Y ⊆ U. The adhesion-lower and adhesion-upper approximations satisfy all the properties (L1 - L10) and (U1 - U10).

Proof. The properties (L1)-(L3), (U1)-(U3), (L5), (L6), (U5), and (U6) are obvious. So, we will prove the other properties as follows:

(L4) Firstly, since X ∩ Y ⊆ X and X ∩ Y ⊆ Y. Then, $\underline{H} (X \cap Y) \subseteq \underline{H} (X)$ and $\underline{H} (X \cap Y) \subseteq \underline{H} (Y)$ . Therefore, $\underline{H} (X \cap Y) \subseteq \underline{H} (X) \cap \underline{H} (Y)$ .

Now let $x \in \underline{H} (X) \cap \underline{H} (Y)$ , then $x \in \underline{H} (X)$ and $x \in \underline{H} (Y)$ , which means that h (x) ⊆ X and h (x) ⊆ Y. Thus h (x) ⊆ X ∩ Y and this implies $x \in \underline{H} (XY)$ . Hence, $\underline{H} (X) \cap \underline{H} (Y) \subseteq \underline{H} (X \cap Y)$ and therefore $\underline{H} (X) \cap \underline{H} (Y) = \underline{H} (X \cap Y)$ .

(U4) similar to (L4).

(L7) By using Definition 3.2, we have: $\begin{matrix} (\bar{H} (X))^{c} = {[{x \in U : h (x) \cap X \neq \emptyset}]}^{c} \\ = {x \in U : h (x) \cap X = \emptyset} . \end{matrix}$

Since x ∈ h (x) , ∀ x ∈ U, then we get: $\begin{matrix} (\bar{H} (X))^{c} = {x \in U : h (x) \cap X = \emptyset} = \\ {x \in U : h (x) \subseteq X^{c}} = \underline{H} (X^{c}) . \end{matrix}$

(U7) similar to (L7).

(L8) Firstly, we have $\underline{H} (\underline{H} (X)) \subseteq \underline{H} (X)$ . Conversely, if $y \in \underline{H} (X)$ , then h (y) ⊆ X. So, we must prove that $h (y) \subseteq \underline{H} (X)$ as follows:

Let z ∈ h (y), then, by Lemma 3.1, h (z) = h (y). Thus h (z) ⊆ X and this implies $z \in \underline{H} (X)$ . Hence, $h (y) \subseteq \underline{H} (X)$ and $y \in \underline{H} (\underline{H} (X))$ . Therefore, $\underline{H} (X) \subseteq \underline{H} (\underline{H} (X))$ and then $\underline{H} (\underline{H} (X)) = \underline{H} (X)$ .

(U8) similar to (L8).

The proof of (L9), (L10), (U9), and (U10), by using Lemmas 3.1, 3.2, and 3.3 is obvious. ■

Corollary 3.1 Consider R is a binary relation on U, and X ⊆ U. The adhesion-lower and adhesion-upper approximations satisfy the following: $\underline{H} (X) = \bar{H} (X)$ if and only $\underline{H} (X) = X$ or $\bar{H} (X) = X$ .

Remark 3.1 In Theorem 3.1, the converse of properties (L6) and (U6) needn’t be true in general as illustrated in Example 3.1.

Example 3.1 If U ={ a, b, c, d } and = {(a, a), (b, b) , (b, c) , (c, c) , (d, a)}. Then we get: $aR = dR = {a}, bR = {b, c} and cR = {c} .$

Thus, the adhesion-neighborhoods generated by this relation are: $h (a) = h (d) = {a, d}, h (b) = {b} and h (c) = {c} .$

Now, let A ={ b, d } and B ={ a, b, c }. Then, A∩ B = { b } and A ∪ B = U.

By using Definition 3.2, we get:

$\underline{H} (A) = {b}$ , $\underline{H} (B) = {b, c}$ and $\underline{H} (A \cup B) = U$ . Obviously, $\underline{H} (A) \cup \underline{H} (B) ⊊ \underline{H} (A \cup B)$ .

Also, $\bar{H} (A) = {a, b, d}$ , $\bar{H} (B) = U$ and $\bar{H} (A \cap B) = {b}$ . Clearly, $\bar{H} (A \cap B) ⊊ \bar{H} (A) \cap \bar{H} (B)$ .

Proposition 3.2 Let R be an equivalence relation on U, and X ⊆ U. Then the adhesion-lower and adhesion-upper approximations satisfy the following:

h (x) = [x] _R, ∀x ∈ U.

$H (U) = U / R$ .

$\underline{H} (X) = \underline{Apr} (X) = {x \in U : {[x]}_{R} \subseteq X}$ .

$\bar{H} (X) = \bar{Apr} (X) = {x \in U : {[x]}_{R} \cap X \neq \emptyset}$ .

Where, $\underline{Apr} (X)$ and $\bar{Apr} (X)$ represent Pawlak’s rough set approximations, respectively.

Proof. Only, the first statement will be proved and the others can be made similarly.

(i) Let R be an equivalence relation on U. Then h (x) = { y ∈ U : yR = xR } = { y ∈ U : [y] _R = [x] _R } = [x] _R. ■

Remark 3.2 From Theorem 3.1 and Proposition 3.2, we conclude that the proposed approximations (the adhesion-approximations) represent a generalization to Pawlak’s rough sets and its generalizations since these approximations induced via a general binary relation and has properties like the properties of Pawlak’s approximations (without any restrictions on the relation). Therefore, we can say that the adhesion-approximations represent the natural extension of Pawlak’s rough set models and then Pawlak’s approach represents an especial case of the proposed approach (adhesion-approximations).

The fundamental aims of the following results are to exemplify the relationships between our approach and the other methods (Yao’s [18] and Allam [19]).

Lemma 3.3 If R is a reflexive relation on U. Then for every w ∈ U:

h (w) ⊆ wR.

wR ⊆ wR.

Proof. Only, the first statement will be proved and the other can be made similarly.

Let s ∈ h (w), then sR = wR. By reflexivity of R, we have s ∈ sR which implies s ∈ wR. ■

Theorem 3.2 If R is a reflexive relation on U. Then for each X ⊆ U:

$\underline{R} (X) \subseteq \underline{H} (X) \subseteq X \subseteq \bar{H} (X) \subseteq \bar{R} (X)$ .

$B nd (X) \subseteq B_{R} (X)$ and δ_R (X) ⩽ λ (X).

If X is an exact set in Yao’s approach, then it is an adhesion-exact set.

$\underline{R} (X) \subseteq L (X) \subseteq X \subseteq \bar{U} (X) \subseteq \bar{R} (X)$ .

$B_{〈 R 〉} (X) \subseteq B_{〈 R 〉} (X)$ and δ_R (X) ⩽ δ_R (X).

If X is an exact set in Yao’s approach, then it is an exact set in Allam’s approach.

Proof. By using Lemma 3.3, the proof is obvious. ■

Remark 3.3

Theorem 3.2 illustrates that our method (reps. Allam method) is more accurate and stronger than Yao’s technique.

If R is a reflexive relation on U, then the suggested method in Definition 3.2 is independent of Allam’s technique [19], that is the two approaches are not comparable in general.

The converse of the above results is not true.

We propose the next example to explain the above remark.

Example 3.2 Consider U ={ a, b, c, d } and the reflexive relation R = {(a, a) , (a, d) , (b, b) , (b, c), (c, a) (c, c) , (d, a) , (d, d)} on U. Then we get the following neighborhoods

Successor neighborhoods: aR = dR ={ a, d }, bR ={ b, c } and cR ={ a, c }.

Minimal neighborhoods: aR ={ a }, bR ={ b, c }, cR ={ c } and dR ={ a, d }.

Adhesion-neighborhoods: h (a) = h (d) ={ a, d }, h (b) ={ b } and h (c) ={ c }.

Now, if A ={ b } and B ={ a, b }. Then, we get

$\underline{H} (A) = \bar{H} (A) = A$ , $B nd (A) = \emptyset$ and λ (A) = 1 which means that A is an exact set. But, by using Yao method, we have $\underline{R} (A) = \emptyset$ , $\bar{R} (A) = {a, c, d}$ , $B_{R} (A) = {a, c, d}$ and δ_R (A) = 0. Thus A is a rough set.

$\underline{H} (B) = {b}$ , $\bar{H} (B) = {a, b, d}$ , $B nd (B) = {a, d}$ and $λ (B) = \frac{1}{3}$ which implies B is an adhesion rough set. But using Allam method, we get $L (A) = \emptyset$ , $U (A) = {b}$ , $B_{〈 R 〉} (A) = {b}$ and δ_〈R〉 (A) = 0 which implies A is a rough set.

Moreover, $L (B) = {a}$ , $U (B) = {a, b, d}$ , $B_{〈 R 〉} (B) = {b, d}$ and $δ_{〈 R 〉} (B) = \frac{1}{3}$ which implies A is a rough set.

The main goal of the following example is to demonstrate that the suggested method in Definition 3.2 represents the best extension of Pawlak’s methodology. Also, we show that our approaches extend the application field because the adhesion-approximations depend basically on a general binary relation, and hence all Pawlak’s properties are held without any restrictions. On the other hand, the previous methods (Yao, and Allam approaches) can’t apply in the case of general relations, and therefore there some problems in the main properties of the approximations as shown in the following example.

Example 3.3 Consider U ={ e₁, e₂, e₃, e₄ } and R = {(e₁, e₁) , (e₁, e₂) , (e₂, e₂) , (e₂, e₃) , (e₃, e₂),

(e₃, e₃)}. Then we get:

e₁R ={ e₁, e₂ }, e₂R ={ e₂, e₃ }, e₃R ={ e₂, e₃ } and e₄R =∅.

Thus, the minimal neighborhoods generated by this relation are:

e₁R ={ e₁, e₂ }, e₂R ={ e₂ }, e₃R ={ e₂, e₃ } and e₄R =∅.

Thus, the adhesion neighborhoods generated by this relation are:

h (e₁) ={ e₁ }, h (e₂) = h (e₃) ={ e₂, e₃ } and

h (e₄) ={ e₄ }.

We will compute the approximations, boundary regions, and the accuracy measures of all subsets of U using the proposed methods in (Definitions 3.2 and 3.3), Yao method (Definition 2.6), and Allam et al. method (Definition 2.8) as shown in Table 3.1.

Table 3.1
Comparison between the proposed approaches and the previous methods (namely, Yao and Allam et al. approaches)

Diagram 3.1 summarizes the relationships between the suggested approach and the other methods (namely, Yao and Allam approaches) in the case of reflexivity of the relation (where each arrow represents ⊆).

Diagram 3.1 The relationships between the current method and Yao [18] and Allam et al. [19] approaches

Note that: from Table 3.1 , we notice that, by using Yao and Allam et al. methods, there are some contradictions to Rough set theory, for example:

The lower approximation of ∅ is not empty (For example, see red shaded cells).

The upper approximation of U is not equal to U (For example, see red shaded cells).

The lower approximation of some subsets is not included in the set and is not included in its upper approximation (For example, see the yellow shaded cells).

4 Generalized nano-topology based on generalized rough sets

For the first time, the notion of Nano topology has been proposed by Thivagar and Richard [49] as a new type of topological spaces, essentially dependent on Pawlak’s approximations and boundary region of a rough set. Here, we extend this concept to generalized rough sets induced by a binary relation. For the first time, the notion of Nano topology has been proposed by Thivagar and Richard [49] as a new type of topological spaces, essentially dependent on Pawlak’s approximations and boundary region of a rough set. Here, we extend this concept to generalized rough sets induced by a binary relation.

Definition 4.1 [49] Consider U to be a finite set, and R be an equivalence relation on U. If $\bar{Apr} (A)$ , $\underline{Apr} (A)$ and BND_R (A) are Pawlak’s rough upper, lower and boundary of a subset A ⊆ U respectively, then the class:

$T_{R} = {U, \emptyset, \underline{Apr} (A), \bar{Apr} (A), {BND}_{R} (A)}$ is a topology on U and it is said to be a nano-topology generated by A.

The following result extends the above definition to any generalized rough sets. In fact, the main goal of it is to illustrate the conditions that any generalized rough sets (Whatever its definition) generate a nano-topology.

Theorem 4.1 Suppose that U be a finite set and $L_{*} (A)$ , $U^{*} (A)$ , and $B (A)$ are generalized lower, upper and boundary of a subset A ⊆ U, respectively. The class $τ^{GN} = {U, \emptyset, L_{*} (A), U^{*} (A), B (A)}$ is a topology on U if $L_{*} (A)$ and $U^{*} (A)$ satisfy $L_{*} (A) \subseteq A \subseteq U^{*} (A)$ , for every A ⊆ U, where $B (A) = U^{*} (A) - L_{*} (A)$ and τ^GN is called a generalized nano-topology (in briefly, GN-topology) on U associated A ⊆ U.

Proof.

It is clear that U and ∅ ∈ τ^GN.

Since $L_{*} (A) \subseteq A \subseteq U^{*} (A)$ , for every A ⊆ U. Then, $L_{*} (A) \cap U^{*} (A) = L_{*} (A)$ , $L_{*} (A) \cap B (A) = \emptyset$ and $U^{*} (A) \cap B (A) = B (A)$ . Therefore, τ^GN is closed under a finite intersection.

In a similar way to (2), we can prove that τ^GN is closed under an arbitrary union.

Accordingly, τ^GN is a topology on U. ■

Corollary 4.1 Let U be a finite set, and A ⊆ U. Then the base of the GN-topology τ^GN is given by the class: $β^{GN} = {U, L_{*} (A), B (A)}$ .

Note that: The above-mentioned theorem presented the conditions of the generation of nano-topology by any generalized rough sets. This theorem, therefore, extends the nano-topology field of application to any binary relation. Therefore, we use it to generate a generalized nano-topology by using adhesion-approximations as a generalized type of nano-topology, as illustrated in the following result.

Theorem 4.2 Let R be any arbitrary binary relation on the finite set U. Then, the class $T^{GN} = {U, \emptyset, \underline{H} (A), \bar{H} (A), B nd (A)}$ is a GN-topology on U associated to A ⊆ U, such that $\underline{H} (A)$ , $\bar{H} (A)$ and $B nd (A)$ are the adhesion-lower, adhesion-upper and adhesion-boundary of A.

Proof. By using Theorem 3.1, $\underline{H} (A) \subseteq A \subseteq \bar{H} (A)$ , for every A ⊆ U. Therefore, $T^{GN}$ is a GN-topology on U with respect to A ⊆ U. ■

Corollary 4.2 Let U be a finite set and A ⊆ U. Then the base of the GN-topology $T^{GN}$ is given by the class: $B^{GN} = {U, \underline{H} (A), B nd (A)}$ .

The main objective of the following theorems is to illustrate the conditions of generating a GN-topology by using Yao and Allam approximations.

Theorem 4.3 Let U be a finite set, R be a binary relation on U and A ⊆ U. Then the class

Yao’s approach: $T_{1}^{GN} = {U, \emptyset, \underline{R} (A), \bar{R} (A), B_{R} (A)}$ is a GN-topology on U if and only if R is a reflexive relation.

Allam’s approach: $T_{2}^{GN} = {U, \emptyset, L (A), U (A), B_{〈 R 〉} (A)}$ a GN-topology on U if and only if R is an inverse serial relation.

Proof.

By using Theorem 2.1, the proof is obvious.

By using Theorem 2.3, the proof is obvious. ■

The next example demonstrates that the above GN-topologies which are generated by using different types of rough approximations, needn’t be comparable (that is, they are independent).

Example 4.1 Consider U ={ a, b, c, d } and R = {(a, a) , (a, b) , (b, b) , (b, a) , (c, a), (c, c) , (d, d)} is a reflexive relation on U. Therefore, we get the following neighborhoods.

Successor neighborhoods: aR = bR ={ a, b }, cR ={ a, c } and dR ={ d }.

Minimal neighborhoods: aR ={ a }, bR ={ a, b }, cR ={ a, c } and dR ={ d }.

Adhesion-neighborhoods: h (a) = h (b) ={ a, b }, h (c) ={ c } and h (d) ={ d }.

Hence, the GN-topologies of a subset X ={ b, c, d } are

Using Yao’s approach: $T_{1}^{GN} = {U, \emptyset, {d}, {a, b, c}} .$

Using Allam’s approach: $T_{2}^{GN} = {U, \emptyset, {d}, {b, c}, {b, c, d}} .$

Using adhesion approach: $T_{3}^{GN} = {U, \emptyset, {c, d}, {a, b}} .$

Obviously, $T_{1}^{GN} ⊊ T_{2}^{GN}$ , $T_{1}^{GN} ⊊ T_{3}^{GN}$ and $T_{2}^{GN} ⊊ T_{3}^{GN}$ . Hence, $T_{1}^{GN}$ , $T_{2}^{GN}$ and $T_{3}^{GN}$ are independent.

5 Medical applications for diagnosis of lung cancer disease

Lung cancer is the basic cause of cancer’s death for both men and women in the Middle-east and worldwide. Cigarette smoking is the main danger factor in the advance of lung cancer. The rate of lung cancer is powerfully associated with cigarette smoking, with about 90% of lung cancers owing to tobacco usage. Inactive experience with tobacco smoke (passive smoking) may also cause lung cancer in non-smokers. In light of this, people are trying to improve specialized lung cancer systems with the help of mathematics.

5.1 Data of medical diagnosis of a lung cancer disease

Example 5.1 As shown in Table 5.1, the data on lung cancer in our approach is entered after the pre-processing phase and is then classified. Classification (using a general binary relation) and inference are based on knowledge-based information. The data for our current work is found from the Respiratory Department of Nanjing Chest Hospital, China. The experimental results are discussed in a preparatory study of 45 patients divided over 30 males (p1, p3, p6, p9, p11, p16, p18, p20, p22, p25, p30, p32, p33, p36, p45) and 15 females. These data contain 6 attributes:

Weight loss (WL).

Shortness of breath (SHB).

Chest pain (CHP).

Blood in sputum (BS).

Persistence coughs (PC).

Age.

Table 5.1
An information system (Decision from specialist doctor) [53]

P WL SHB CHP PC BS Age Decision P WL SHB CHP PC BS Age Decision

P₁ 3.8 38 48 18 45 55 No cancer P₂₄ 4.2 60 49 29 49 66 Lung cancer

P₂ 3.7 40 50 40 44 60 Lung cancer P₂₅ 4.9 61 55 37 48 64 Lung cancer

P₃ 5 50 49 33 50 49 No cancer P₂₆ 5.1 44 56 49 44 49 No cancer

P₄ 4 44 55 45 54 45 No cancer P₂₇ 4.5 43 51 43 47 46 No cancer

P₅ 4.1 38 60 48 49 48 No cancer P₂₈ 5.7 39 57 44 45 54 No cancer

P₆ 4.5 54 51 35 52 50 No cancer P₂₉ 6 38 49 53 44 56 Lung cancer

P₇ 3.7 59 56 39 53 54 No cancer P₃₀ 3.6 43 48 50 54 74 Lung cancer

P₈ 3.6 62 60 29 45 66 Lung cancer P₃₁ 3.8 41 47 28 49 71 No cancer

P₉ 3.9 63 47 44 54 70 Lung cancer P₃₂ 3.9 42 60 59 50 46 No cancer

P₁₀ 4.4 55 53 57 47 55 Lung cancer P₃₃ 3.8 55 54 48 51 51 Lung cancer

P₁₁ 4.8 40 59 50 46 62 Lung cancer P₃₄ 3.9 53 47 55 49 56 Lung cancer

P₁₂ 4.6 60 57 59 50 72 Lung cancer P₃₅ 4.7 54 49 56 53 55 Lung cancer

P₁₃ 3.7 37 47 16 44 45 No cancer P₃₆ 4.8 63 55 18 47 62 Lung cancer

P₁₄ 3.6 39 50 19 50 61 Lung cancer P₃₇ 4.9 62 52 21 54 61 Lung cancer

P₁₅ 4.1 42 51 30 47 59 Lung cancer P₃₈ 5.1 46 54 25 49 63 No cancer

P₁₆ 4.7 44 47 32 53 70 Lung cancer P₃₉ 5.3 47 56 16 50 45 No cancer

P₁₇ 5 45 49 40 54 48 No cancer P₄₀ 5.2 49 57 26 51 49 No cancer

P₁₈ 3.7 50 54 42 45 49 No cancer P₄₁ 5.5 44 60 25 52 75 Lung cancer

P₁₉ 3.9 55 52 16 47 50 No cancer P₄₂ 3.7 43 49 39 54 56 No cancer

P₂₀ 4 52 53 20 51 72 Lung cancer P₄₃ 3.9 40 49 52 48 65 No cancer

P₂₁ 4.1 53 60 22 53 73 Lung cancer P₄₄ 4.1 57 55 54 50 55 Lung cancer

P₂₂ 4.7 38 54 24 52 55 Lung cancer P₄₅ 4.3 63 56 15 54 59 Lung cancer

P₂₃ 4.5 40 48 27 50 60 Lung cancer

P	WL	SHB	CHP	PC	BS	Age	Decision	P	WL	SHB	CHP	PC	BS	Age	Decision
P₁	3.8	38	48	18	45	55	No cancer	P₂₄	4.2	60	49	29	49	66	Lung cancer
P₂	3.7	40	50	40	44	60	Lung cancer	P₂₅	4.9	61	55	37	48	64	Lung cancer
P₃	5	50	49	33	50	49	No cancer	P₂₆	5.1	44	56	49	44	49	No cancer
P₄	4	44	55	45	54	45	No cancer	P₂₇	4.5	43	51	43	47	46	No cancer
P₅	4.1	38	60	48	49	48	No cancer	P₂₈	5.7	39	57	44	45	54	No cancer
P₆	4.5	54	51	35	52	50	No cancer	P₂₉	6	38	49	53	44	56	Lung cancer
P₇	3.7	59	56	39	53	54	No cancer	P₃₀	3.6	43	48	50	54	74	Lung cancer
P₈	3.6	62	60	29	45	66	Lung cancer	P₃₁	3.8	41	47	28	49	71	No cancer
P₉	3.9	63	47	44	54	70	Lung cancer	P₃₂	3.9	42	60	59	50	46	No cancer
P₁₀	4.4	55	53	57	47	55	Lung cancer	P₃₃	3.8	55	54	48	51	51	Lung cancer
P₁₁	4.8	40	59	50	46	62	Lung cancer	P₃₄	3.9	53	47	55	49	56	Lung cancer
P₁₂	4.6	60	57	59	50	72	Lung cancer	P₃₅	4.7	54	49	56	53	55	Lung cancer
P₁₃	3.7	37	47	16	44	45	No cancer	P₃₆	4.8	63	55	18	47	62	Lung cancer
P₁₄	3.6	39	50	19	50	61	Lung cancer	P₃₇	4.9	62	52	21	54	61	Lung cancer
P₁₅	4.1	42	51	30	47	59	Lung cancer	P₃₈	5.1	46	54	25	49	63	No cancer
P₁₆	4.7	44	47	32	53	70	Lung cancer	P₃₉	5.3	47	56	16	50	45	No cancer
P₁₇	5	45	49	40	54	48	No cancer	P₄₀	5.2	49	57	26	51	49	No cancer
P₁₈	3.7	50	54	42	45	49	No cancer	P₄₁	5.5	44	60	25	52	75	Lung cancer
P₁₉	3.9	55	52	16	47	50	No cancer	P₄₂	3.7	43	49	39	54	56	No cancer
P₂₀	4	52	53	20	51	72	Lung cancer	P₄₃	3.9	40	49	52	48	65	No cancer
P₂₁	4.1	53	60	22	53	73	Lung cancer	P₄₄	4.1	57	55	54	50	55	Lung cancer
P₂₂	4.7	38	54	24	52	55	Lung cancer	P₄₅	4.3	63	56	15	54	59	Lung cancer
P₂₃	4.5	40	48	27	50	60	Lung cancer

5.2 Decision-making via adhesion-approximations

Decision-making plays a vital role in our daily lives, and this process provides the best alternative among the different choices. In this subsection, we illustrate the importance of adhesion-approximations in decision-making which help the doctor in the medical diagnosis of lung cancer. In fact, we analyze the data in Table 5.1 and calculate the approximations of certain subsets (namely lung cancer patients) using the proposed method and the previous approaches [18] and [19]. In this application, we use a general binary relation to illustrate the significance of the suggested technique in decision-making. Therefore, we prove that Yao and Allam’s methods can’t apply here and hence we can say that our method extends the application field of rough sets. Accordingly, we demonstrate that the suggested tools are more accurate than the other methods.

Consider the following binary relation R_J on the universe U ={ p₁, p₂, p₃, …, p₄₅ }:

For each x, y ∈ U, xR_Jy ⇔ v_J (x) < v_J (y), where J∈ { WL, SHB, CHP, PC, BS, Age }.

–The successor neighborhoods of x ∈ U (that generated from Table 5.1) are: $\begin{matrix} p_{1} R = {p_{11}, p_{12}, p_{15}, p_{20}, p_{21}, p_{24}, p_{25}, p_{37}, p_{38}, p_{41}, p_{43}}, \\ p_{2} R = p_{10} R = p_{31} R = p_{34} R = p_{43} R = {p_{12}}, \end{matrix}$ $\begin{matrix} p_{3} R = \dots = p_{9} R = p_{11} R = p_{12} R = p_{16} R = \\ p_{17} R = p_{21} R = \dots = p_{30} R = p_{32} R = p_{33} R = \\ p_{35} R = \dots = p_{41} R = p_{42} R = p_{44} R = p_{45} R = \emptyset, \\ p_{13} R = {p_{1}, p_{3}, p_{5}, p_{6}, p_{10}, p_{11}, p_{12}, p_{15}, p_{16}, p_{17}, \\ p_{20}, \dots, p_{25}, p_{27}, p_{28}, p_{31}, \dots, p_{38}, p_{40}, p_{41}, p_{43}, p_{44}} \\ p_{14} R = {p_{20}, p_{21}, p_{41}}, p_{15} R = {p_{12}, p_{25}}, p_{15} R = {p_{12}, p_{25}} \\ p_{18} R = {p_{12}, p_{44}}, p_{19} R = {p_{12}, p_{25}, p_{44}} and \\ p_{20} R = {p_{21}} . \end{matrix}$

–The minimal neighborhoods of x ∈ U (that generated from Table 5.1) are: $\begin{matrix} 〈 p_{1} 〉 R = 〈 p_{3} 〉 R = 〈 p_{5} 〉 R = 〈 p_{6} 〉 R = 〈 p_{10} 〉 R = \\ 〈 p_{16} 〉 R = 〈 p_{17} 〉 R = 〈 p_{23} 〉 R = 〈 p_{27} 〉 R = 〈 p_{28} 〉 R \\ = 〈 p_{31} 〉 R = \dots = 〈 p_{36} 〉 R = 〈 p_{40} 〉 R = {p_{1}, p_{3}, \\ p_{5}, p_{6}, p_{10}, p_{11}, p_{12}, p_{15}, p_{16}, p_{17}, p_{20}, \dots, p_{25}, \\ p_{27}, p_{28}, p_{31}, \dots, p_{38}, p_{40}, p_{41}, p_{43}, p_{44}}, \\ 〈 p_{2} 〉 R = 〈 p_{4} 〉 R = 〈 p_{7} 〉 R = 〈 p_{8} 〉 R = 〈 p_{9} 〉 R = \\ 〈 p_{13} 〉 R = 〈 p_{14} 〉 R = 〈 p_{18} 〉 R = 〈 p_{19} 〉 R = 〈 p_{26} 〉 R \\ = 〈 p_{29} 〉 R = 〈 p_{30} 〉 R = 〈 p_{39} 〉 R = 〈 p_{42} 〉 R = \\ 〈 p_{45} 〉 R = \emptyset, \end{matrix}$ $\begin{matrix} 〈 p_{11} 〉 R = 〈 p_{15} 〉 R = 〈 p_{20} 〉 R = 〈 p_{22} 〉 R = 〈 p_{24} 〉 R \\ = 〈 p_{37} 〉 R = 〈 p_{38} 〉 R = 〈 p_{43} 〉 = {p_{11}, p_{12}, p_{15}, \\ p_{20}, p_{21}, p_{24}, p_{25}, p_{37}, p_{38}, p_{41}, p_{43}} \\ 〈 p_{12} 〉 R = {p_{12}}, 〈 p_{21} 〉 R = {p_{21}} \\ 〈 p_{25} 〉 R = {p_{12}, p_{25}}, 〈 p_{41} 〉 R = {p_{20}, p_{21}, p_{41}}, \\ ; and 〈 p_{44} 〉 R = {p_{12}, p_{44}} . \end{matrix}$

–The adhesion-neighborhoods of x ∈ U (that generated from Table 5.1) are: $\begin{matrix} h (p_{1}) = {p_{1}}, h (p_{13}) = {p_{13}}, h (p_{14}) = {p_{14}}, \\ h (p_{15}) = {p_{15}}, h (p_{18}) = {p_{18}}, \\ h (p_{19}) = {p_{19}}, h (p_{20}) = {p_{20}}, \\ h (p_{2}) = h (p_{10}) = h (p_{31}) = h (p_{34}) = \\ h (p_{43}) = {p_{2}, p_{10}, p_{31}, p_{34}, p_{43}}, and \\ h (p_{3}) = \dots = h (p_{12}) = h (p_{16}) = h (p_{17}) = \\ h (p_{21}) = \dots = h (p_{30}) = h (p_{32}) = h (p_{33}) = \\ h (p_{36}) = h (p_{38}) = \dots = h (p_{42}) = h (p_{44}) = \\ h (p_{45}) = {p_{3}, \dots, p_{12}, p_{16}, p_{17}, p_{21}, \dots, p_{30}, p_{32}, \\ p_{33}, p_{35}, p_{36}, p_{38}, \dots, p_{42}, p_{44}, p_{45}} . \end{matrix}$

Now, we will compute the approximations of two subsets, first the set of patients, which have no lung cancer and the second subset is the set of lung cancer patients. And then, we compare among different methods of approximations (the proposed method and the other methods).

•Patients have no lung cancer: $\begin{matrix} X = {p_{1}, p_{3}, \dots, p_{7}, p_{13}, p_{17}, p_{18}, p_{19}, p_{26}, p_{27}, p_{28}, \\ p_{31}, p_{32}, p_{38}, p_{39}, p_{40}, p_{42}, p_{43}} . \end{matrix}$

Yaoapproach [18]: $\begin{matrix} \underline{R} (X) = {p_{3}, \dots, p_{9}, p_{11}, p_{12}, p_{16}, p_{17}, p_{21}, \dots, p_{30}, p_{32}, \\ p_{33}, p_{35}, \dots, p_{42}, p_{44}, p_{45}} and \bar{R} (X) = {p_{1}, p_{13}} . \end{matrix}$

Note that: $\underline{R} (X) ⊊ X ⊊ \bar{R} (X)$ and thus X is a totally rough (un-definable) set. Moreover, there are several elements belong to $\underline{R} (X)$ , but they are not members in X which is a contradiction to Pawlak’s rough sets assumption.

Allam et al. approach [19]: $\begin{matrix} L (X) = {p_{2}, p_{4}, p_{7}, p_{8}, p_{9}, p_{13}, p_{14}, p_{18}, p_{19}, p_{26}, p_{29}, p_{30}, \\ p_{39}, p_{42}, p_{45}} \\ and \\ U (X) = {p_{3}, p_{5}, p_{7}, p_{10}, p_{11}, p_{15}, p_{17}, p_{22}, p_{24}, p_{27}, p_{28}, p_{32}, \\ p_{33}, p_{35}, p_{40}, p_{43}} . \end{matrix}$

Note that: $L (X) ⊊ X ⊊ U (X)$ and thus X is a totally rough (un-definable) set. Moreover, there are several elements belong to $L (X)$ , but they are not members in X which is a contradiction to Pawlak’s methodology.

Adhesion approach (Current method): $\begin{matrix} \underline{H} (X) = {p_{1}, p_{13}, p_{18}, p_{19}}, and \bar{H} (X) \\ = U - {p_{2}, p_{10}, p_{14}, p_{15}, p_{20}, p_{22}, p_{31}, p_{34}} . \end{matrix}$

Note that: $\underline{H} (X) \subseteq X \subseteq \bar{H} (X)$ which interprets that the patients {p₁, p₁₃, p₁₈, p₁₉} surely have lung cancer and the patients U - {p₁, p₂, p₁₀, p₁₃, p₁₅, p₁₈, p₂₀, p₂₂, p₃₁, p₃₄} possibly have lung cancer. Moreover, we able to decide the patients, which have no lung cancer by the set $N eg (X) = U - \bar{H} (X)$ . These results assert and agree with the decisions in the decision system in Table 5.1.

Thus, we can say that the proposed approximations represent important tools for decision-making in real-life problems and more accurate than other methods (such as [18] and [19]).

Similarly, we can compare the proposed method and the previous methods, by computing the approximations of the set (Y = U - X) of patients that have no lung cancer disease. And thus, we obtain the same results.

Decision-making achieves a critical role in our daily life, and this procedure produces the finest alternate amongst different choices. So, we present Algorithm 5.1, for the decision-making of an information system in terms of the adhesion-approximations.

Algorithm 5.1

Step 1: Input the universe set U and set AT ={ a₁, a₂, a₃, …, a_n } to attributes that represent data as an information table, rows of which are categorized by the attributes AT, columns by objects and table entries are attribute values.

Step 2: Input the binary relation R by the expert and thus compute the neighborhoods of all elements in U by using Definition 2.5.

Step 3: Compute the adhesion-neighborhoods of each element in U by using Definition 3.1.

Step 4: Compute the adhesion-upper approximation, say $\bar{H} (X)$ , and the adhesion-lower approximation, say, $\underline{H} (X)$ , according to Definition 3.2.

Step 5: Determine adhesion-boundary, say $B nd (X)$ , from Step 4, for every X ⊆ U, using Definition 3.3.

Step 6: Calculate the accuracy of the approximation, say λ (X), from Step 4, for every X ⊆ U, using Definition 3.3.

Step 7: Decide, exactly, rough sets and exact sets.

5.3 Topological reduction of attributes

In the following discussion, we will apply the suggested methods to make a topological reduction for the attributes of Table 5.1. Therefore, we identify the most risk factors for cause lung cancer disease. In fact, we use the new notion “generalized nano-topology” to identify these factors by obtaining the core of attributes by using a topological reduction of attributes of Table 5.1.

Here, we do a topological reduction of the attributes for “female’s patients” only, and similarly, we can apply this method for “male’s patients” and also for all patients in Table 5.1. Firstly, we get Table 5.2 which represents the information system of female patients.

Table 5.2
The information system of female’s patients

WL SHB CHP PC BS Age Decision

p ₂ 03.7 40 50 40 44 60 Yes

p ₄ 04 44 55 45 54 45 No

p ₅ 04.1 38 60 48 49 48 No

p ₁₀ 04.4 55 53 57 47 55 Yes

p ₁₂ 04.6 60 57 59 50 72 Yes

p ₁₃ 03.7 37 47 16 44 45 No

p ₁₄ 03.6 39 50 19 50 61 Yes

p ₁₅ 04.1 42 51 30 47 59 Yes

p ₁₇ 05 45 49 40 54 48 No

p ₂₁ 04.1 53 60 22 53 73 Yes

p ₂₃ 04.5 40 48 27 50 60 Yes

p ₂₄ 04.2 60 49 29 49 66 Yes

p ₃₁ 03.8 41 47 28 49 71 No

p ₃₄ 03.9 53 47 55 49 56 Yes

p ₃₅ 04.7 54 49 56 53 55 Yes

	WL	SHB	CHP	PC	BS	Age	Decision
p ₂	03.7	40	50	40	44	60	Yes
p ₄	04	44	55	45	54	45	No
p ₅	04.1	38	60	48	49	48	No
p ₁₀	04.4	55	53	57	47	55	Yes
p ₁₂	04.6	60	57	59	50	72	Yes
p ₁₃	03.7	37	47	16	44	45	No
p ₁₄	03.6	39	50	19	50	61	Yes
p ₁₅	04.1	42	51	30	47	59	Yes
p ₁₇	05	45	49	40	54	48	No
p ₂₁	04.1	53	60	22	53	73	Yes
p ₂₃	04.5	40	48	27	50	60	Yes
p ₂₄	04.2	60	49	29	49	66	Yes
p ₃₁	03.8	41	47	28	49	71	No
p ₃₄	03.9	53	47	55	49	56	Yes
p ₃₅	04.7	54	49	56	53	55	Yes

From Table 5.2 , we get

The set of female’s patients is: $\begin{matrix} P = {p_{2}, p_{4}, p_{5}, p_{10}, p_{12}, p_{15}, p_{17}, \\ p_{21}, p_{23}, p_{24}, p_{31}, p_{34}, p_{35}} . \end{matrix}$

The set of attributes is: AT ={ WL, SHB, CHP, PC, BS, Age } and Decision attribute.

Now, to calculate the adhesion-neighborhoods, we define the relation in each issue, according to the requirements of the expert by the binary relation of the objects (set of female’s patients) as follows:

For each x, y ∈ P, xR_Jy ⇔ v_J (x) < v_J (y), where J∈ { WL, SHB, CHP, PC, BS, Age }.

Therefore, all adhesion-neighborhoods for all elements in P are: $\begin{matrix} h (p_{2}) = h (p_{10}) = h (p_{15}) = h (p_{31}) = h (p_{34}) \\ = {p_{2}, p_{10}, p_{15}, p_{31}, p_{34}}, \\ h (p_{4}) = h (p_{5}) = h (p_{12}) = h (p_{17}) = h (p_{21}) \\ = h (p_{23}) = h (p_{24}) = h (p_{35}) = {p_{4}, p_{5}, p_{12}, \\ p_{17}, p_{21}, p_{23}, p_{24}, p_{35}}, h (p_{13}) = {p_{13}} and \\ h (p_{14}) = {p_{14}} . \end{matrix}$

Based on Theorem 4.1 and Corollary 4.1, we calculate the GN-topology of decision-making for two groups of patients:

Case 1: (Patients infected with lung cancer disease)

The set of infected patients with lung cancer disease is X ={ p₂, p₁₀, p₁₂, p₁₄, p₁₅, p₂₁, p₂₃, p₂₄, p₃₄, p₃₅ }.

Therefore, we get $\begin{matrix} \underline{H} (X) = {p_{14}}, \bar{H} (X) = P - {p_{13}} and \\ B nd (X) = P - {p_{13}, p_{14}} . \end{matrix}$

Accordingly, the GN-topology of X, induced by all attributes, is given by: $T_{AT}^{GN} = {P, \emptyset, {p_{14}}, P - {p_{13}}, P - {p_{13}, p_{14}}}$

and its base is $B_{AT}^{GN} = {P, {p_{14}}, P - {p_{13}, p_{14}}}$ .

Step 1: When the attribute (WL) is removed

The adhesion-neighborhoods in this case are: $\begin{matrix} h (p_{2}) = h (p_{10}) = h (p_{15}) = h (p_{31}) = h (p_{34}) \\ = {p_{2}, p_{10}, p_{15}, p_{31}, p_{34}}, \\ h (p_{4}) = h (p_{5}) = h (p_{12}) = h (p_{17}) = h (p_{21}) = \\ h (p_{23}) = h (p_{24}) = h (p_{35}) = {p_{4}, p_{5}, p_{12}, p_{17}, \\ p_{21}, p_{23}, p_{24}, p_{35}}, h (p_{13}) = {p_{13}} and \\ h (p_{14}) = {p_{14}} . \end{matrix}$

Consequently, we get $\begin{matrix} \underline{H} (X) = {p_{14}}, \bar{H} (X) = P - {p_{13}} and \\ B nd (X) = P - {p_{13}, p_{14}} . \end{matrix}$

Thus, the GN-topology of X is given by $\begin{matrix} T_{AT - WL}^{GN} = {P, \emptyset, {p_{14}}, P - {p_{13}}, P - {p_{13}, p_{14}}} \\ = T_{AT}^{GN} . \end{matrix}$

The base is given by $B_{AT - WL}^{GN} = {P, {p_{14}}, P - {p_{13}, p_{14}}} = B_{AT}^{GN} .$

Step 2: When the attribute (SHB) is removed, by the same manner as Step 1, we get

$\underline{H} (X) = {p_{14}}$ , $\bar{H} (X) = P - {p_{13}}$ and $B nd (X) = P - {p_{13}, p_{14}}$ .

Hence, the GN-topology of X is given by $T_{AT - SHB}^{GN} = {P, \emptyset, {p_{14}}, P - {p_{13}}, P - {p_{13}, p_{14}}}$

$= T_{AT}^{GN}$ . Also, the base is given by $B_{AT - SHB}^{GN} = {P, {p_{14}}, P - {p_{13}, p_{14}}} = B_{AT}^{GN} .$

Step 3: When the attribute (CHP) is removed, by the same manner as Step 1, we get

$\underline{H} (X) = {p_{14}}$ , $\bar{H} (X) = P - {p_{5}, p_{13}}$ and

$B nd (X) = P - {p_{5}, p_{13}, p_{14}}$ .

Thus, the GN-topology of X is given by

$T_{AT - CHP}^{GN} = {P, \emptyset, {p_{14}}, P - {p_{5}, p_{13}},$

$P - {p_{5}, p_{13}, p_{14}}} \neq T_{AT}^{GN}$ .

Also, the base is given by

$B_{AT - CHP}^{GN} = {P, {p_{14}}, P - {p_{5}, p_{13}, p_{14}}} \neq B_{AT}^{GN}$ .

Step 4: When the attribute (BS) is removed, by the same manner as Step 1, we get

$\underline{H} (X) = {p_{2}, p_{10}, p_{14}, p_{15}, p_{34}}$ ,

$\bar{H} (X) = P - {p_{13}, p_{31}}$ and

$B nd (X) = {p_{4}, p_{5}, p_{12}, p_{17}, p_{21}, p_{23}, p_{24}, p_{35}}$ .

Hence, the GN-topology of X is given by

$T_{AT - BS}^{GN} = {P, \emptyset, {p_{2}, p_{10}, p_{14}, p_{15}, p_{34}},$

P -{ p₁₃, p₃₁ } , { p₄, p₅, p₁₂, p₁₇, p₂₁, p₂₃, p₂₄, p₃₅ }}

$\neq T_{AT}^{GN}$ . Also, the base is given by

$B_{AT - BS}^{GN} = {P, {p_{2}, p_{10}, p_{14}, p_{15}, p_{34}}, {p_{4}, p_{5}, p_{12},$

$p_{17}, p_{21}, p_{23}, p_{24}, p_{35}} \neq B_{AT}^{GN}$ .

Step 5: When the attribute (PC) is removed, by the same manner as Step 1, we get

$\underline{H} (X) = {p_{14}}$ , $\bar{H} (X) = P - {p_{13}}$ and $B nd (X) = P - {p_{13}, p_{14}}$ .

Hence, the GN-topology of X is given by:

$\begin{matrix} T_{AT - PC}^{GN} = {P, \emptyset, {p_{14}}, P - {p_{13}}, \\ P - {p_{13}, p_{14}}} = T_{AT}^{GN} . \end{matrix}$

Also, the base is given by: $B_{AT - PC}^{GN} = {P, {p_{14}}, P - {p_{13}, p_{14}}} = B_{AT}^{GN} .$

Step 6: When the attribute (Age) is removed, by the same manner as Step 1, we get

$\underline{H} (X) = {p_{2}, p_{14}}$ , $\bar{H} (X) = P - {p_{13}}$ and

$B nd (X) = P - {p_{2}, p_{13}, p_{14}}$ .

Thus, the GN-topology of X is given by: $\begin{matrix} T_{AT - BS}^{GN} = {P, \emptyset, {p_{2}, p_{14}}, P - {p_{2}, p_{13}, p_{14}}, \\ P - {p_{13}}} \neq T_{AT}^{GN} . \end{matrix}$

Also, the base is given by: $\begin{matrix} B_{AT - BS}^{GN} = {P, {p_{2}, p_{14}}, \\ P - {p_{2}, p_{13}, p_{14}}} \neq B_{AT}^{GN} . \end{matrix}$

Hence, the attributes {CHP, BS, Age} are not dispensable attributes, and {WL, SHB, PC} are dispensable. Consequently, {CHP, BS, Age} is reducing for the information system in (Table 5.2). Hence, the CORE is {CHP, BS, Age} which represents the impact factors for lung cancer disease.

Similarly, we can reduce the attributes for the second group of patients who had no lung cancer disease.

At the end of the paper, we give an algorithm that can be used to make a Topological reduction of attributes for information systems in terms of the adhesion-approximations and GN-topology.

Algorithm 5.2

Step 1: Input the universe set U and set AT ={ a₁, a₂, a₃, …, a_n } to attributes that represent data as an information table, rows of which are characterized by objects, columns by the attributes AT and table entries are attribute values.

Step 2: Compute the adhesion-upper approximation $\bar{H} (X)$ , adhesion-lower approximation, $\underline{H} (X)$ , and adhesion-boundary, $B nd (X)$ , for the decision-set X ⊆ U, using Definitions 3.2 & 3.3.

Step 3: Generate the base $B_{AT}^{GN}$ of GN-topology $T_{AT}^{GN}$ generated by adhesion-approximations using Theorem 4.1 and Corollary 4.1.

Step 4: Eliminate an attribute a₁ from conditions attributes (AT) and find the adhesion-upper approximation, adhesion-lower approximation and adhesion-boundary for the decision-set X on AT –(a₁).

Step 5: Generate the base $B_{AT - a_{1}}^{GN}$ of GN-topology $T_{AT - a_{1}}^{GN}$ generated by adhesion-approximations given in Step 4 using Theorem 4.1 and Corollary 4.1.

Step 6: Repeat the steps 4 and 5 for all of the attributes in (AT).

Step 7: The attributes in (AT) for which $B_{AT - a_{1}}^{GN} \neq B_{AT}^{GN}$ form CORE (U).

6 Conclusion

The main contribution of this research is to present generalized rough sets based on the notion of adhesion-neighborhood, which was used in different forms by [50 –52]. Moreover, new properties and results for adhesion-neighborhoods were illustrated and the suggested approximations have been explained. Comparisons between the proposed approaches (adhesion-approximation) and the previous methods were examined. Besides, Theorem 3.1 and its results have shown that the suggested approaches satisfy all the characteristics of Pawlak’s rough sets without any additional conditions. As an important and crucial goal, generalized nano-topology was proposed, for the first time, as a generalization of the nano-topology [49] generated by generalized rough sets. Theorem 4.1 explained the conditions that the generalized rough sets form a generalized nano-topology and thus Theorem 4.2 demonstrated the conditions that Allam’s approaches and Yao’s approaches generate generalized nano-topology. New links between Topology and Generalized Rough Sets were being superimposed. These links will be useful in other uncertainty representation frameworks, such as three-way decisions.

Finally, we applied the proposed methods in the context of a medical application that was generated by a general binary relation, and thus one can use these approaches in different cases of the relation (such as, reflexive, pre-order, equivalence, etc). The proposed medical application of the diagnosis of pulmonary cancer depends basically on a general binary relation to illustrate the importance of the suggested methods in decision-making problems. So, Pawlak, Yao [18], and Allam [19] methods can’t apply here and hence we can say that our method extends the application field of rough sets. As a realistic application, we used real-life data collected from an experiment involving forty-five patients undergoing treatment at the Nanjing Chest Hospital Respiratory Department, China [53]. Besides, we have proposed a topological reduction, using the suggested methods, to identify the most risk factors for lung cancer. Two algorithms were proposed that can be used for decision-making problems in any information system via our methods.

Footnotes

Acknowledgments

The authors sincerely thank the reviewers for the careful reading and thoughtful comments. The present version of the paper owes much to their precise and valuable remarks that helped in improving the paper.

Compliance with ethical standards

Conflict of interest

All authors declare that there is no conflict of interest regarding the publication of this manuscript.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Availability of data and material

Not applicable.

Funding

Not applicable.

References

Pawlak

, Rough sets, International Journal of Computer & Information Sciences volume 11 (1982), 341–356.

Pawlak

, Rough sets, Theoretical Aspects of Reasoning about Data, Kluwer Acadmic Publishers

Dordrecht, (1991).

Skowron

and Dutta

, Rough sets: past, present, and future, Natural Computing 17(4) (2018), 855–876.

Liang

, Yang

, Chen

and Li

, Stable attribute reduction for neighborhood rough set, Filomat 32(5) (2018), 1809–1815.

Yao

Y.Y.

, Three-way decision and granular computing, International Journal of Approximate Reasoning 103 (2018), 107–123.

El-Sayed

, Al Qubati

A.Q.

and El-Bably

M.K.

, Soft pre-rough sets and its applications in decision making, Mathematical Biosciences and Engineering 17(5) (2020), 6045–6063.

Sharma

H.K.

, Kumari

and Kar

, A rough set approach for forecasting models, Decision Making: Applications in Management and Engineering 3(1) (2020), 1–21.

Karavidic

and Projovic

, A multicriteria decision-making (MCDM) model in the security forces operations based on rough sets, Applications in Management and Engineering 1(1) (2018), 97–120.

Sinani

, Erceg

and Vasiljevic

, An evaluation of a third-party logistics provider: The application of the rough Dombi-Hamy mean operator, Decision Making: Applications in Management and Engineering 3(1) (2020), 92–107.

10.

, Zhan

and Xu

, A novel decision-making approach based on three-way decisions in fuzzy information systems, Information Sciences 541 (2020), 362–390.

11.

, Zhan

, Sun

, Carlos

and Alcantud

, Novel classes of coverings based multigranulation fuzzy rough sets and corresponding applications to multiple attribute group decision-making, Artificial Intelligence Review 53 (2020), 6197–6256.

12.

Carlos

, Alcantud

and Zhan

, Multi-granular soft rough covering sets, Soft Computing 24 (2020), 9391–9402.

13.

El Sayed

, El Safty

M.A.

and El-Bably

M.K.

, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Mathematics 6(7) (2021), 7872–7894.

14.

Abo-Tabl

E.A.

, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Information Sciences 181 (2011), 2587–2596.

15.

Abo-Tabl

E.A.

, Rough sets and topological spaces based on similarity, International Journal of Machine Learning and Cybernetics 4 (2013), 451–458.

16.

Kin

, Yang

and Pei

, Generalized rough sets based on reflexive and transitive relations, Information Sciences 178 (2008), 4138–4141.

17.

Kondo

, On the structure of generalized rough sets, Information Sciences 176 (2006), 589–600.

18.

Yao

Y.Y.

, Generalized rough set models, Rough Sets in knowledge Discovery 1, L. Polkowski, A. Skowron (Eds.), Physica Verlag, Heidelberg (1998), 286–318.

19.

Allam

A.A.

, Bakeir

M.Y.

and Abo-Tabl

E.A.

, New approach for basic rough set concepts. Lecture Notes in Artificial Intelligence 3641, Springer, Regina, (2005), 64–73.

20.

Ali

M.I.

, Davvaz

and Shabir

, Some properties of generalized rough sets, Information Sciences 224 (2013), 170–179.

21.

Abo Khadra

A.A.

, Taher

B.M.

and El-Bably

M.K.

, Generalization of Pawlak approximation space, (Cairo - Volume (), Proceeding of The International Conference on Mathematics: Trends and Developments, The Egyptian Mathematical Society 3 (2007), 335–346.

22.

Abd El-Monsef

M.E.

, Embaby

O.A.

and El-Bably

M.K.

, Comparison between rough set approxi-mations based on different topologies, International Journal of Granular Computing, Rough Sets and Intelligent Systems 3(4) (2014), 292–305.

23.

Zhu

, Topological approaches to covering rough sets, Information Sciences 177(6) (2007), 1499–1508.

24.

Abd El-Monsef

M.E.

, Kozae

A.M.

and El-Bably

M.K.

, On generalizing covering approximation space, Journal of the Egyptian Mathematical Society 23(3) (2015), 535–545.

25.

Abo-Tabl

E.A.

, Topological approaches to generalized definitions of rough multiset approximations, International Journal of Machine Learning and Cybernetics 6 (2015), 399–407.

26.

Al-shami

T.M.

, An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application, Information Sciences 569 (2021), 110–124.

27.

Al-shami

T.M.

, Fu

W.Q.

and Abo-Tabl

E.A.

, New rough approximations based on E-neighborhoods, Complexity, Volume 2021, Article ID 6666853, 6 pages. DOI: https://doi.org/10.1155/2021/6666853.

28.

Salama

A.S.

, Mhemdi

, Elbarbary

O.G.

and Al-shami

T.M.

, Topological approaches for rough continuous functions with applications, Complexity, Volume 2021, Article ID 5586187, 12 pages. DOI: https://doi.org/10.1155/2021/5586187.

29.

, Li

, Huang

, Dai

and Zhang

, A new rough set model based on multi-scale covering, International Journal of Machine Learning and Cybernetics 12 (2021), 243–256.

30.

Kondo

, Note on topologies induced by coverings of approximation spaces, International Journal of Approximate Reasoning 129 (2021), 41–48.

31.

Kandil

, El-Sheikh

S.A.

, Hosny

and Raafat

, Generalization of nano topological spaces induced by different neighborhoods based on ideals and its applications, Tbilisi Math J 14(1) (2021), 135–148.

32.

El-Bably

M.K.

and El Atik

, Soft β-rough sets and their application to determine COVID-19, Turkish Journal of Mathematics 45(3) (2021), 1133–1148.

33.

El-Bably

M.K.

, Fleifel

K.K.

and Embaby

O.A.

, Topological approaches to rough approximations based on closure operators, Granular Computing (2021). DOI:10.1007/s41066-020-00247-x

34.

Zhan

and Wang

, Certain types of soft coverings based rough sets with applications, International Journal of Machine Learning and Cybernetics 10(5) (2019), 1065–1076.

35.

Zhan

, Carlos

and Alcantud

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review 52(4) (2019), 2381–2410.

36.

Abo-Tabl

E.A.

, Topological structure of generalized rough multisets, Life Science Journal 11(4) (2014), 290–299.

37.

El-Bably

M.K.

and Fleifel

K.K.

, Some topological structures by relations, Journal of Computational and Theoretical Nanoscience 14(8) (2017), 4100–4103.

38.

Amer

W.S.

, Abbas

M.I.

and El-Bably

M.K.

, On-near concepts in rough sets with some applications, Journal of Intelligent and Fuzzy Systems 32(1) (2017), 1089–1099.

39.

El-Bably

M.K.

, Comparisons between near open sets and rough approximations, International Journal of Granular Computing, Rough Sets and Intelligent Systems 4(1) (2015), 64–83.

40.

Abd El-Monsef

M.E.

, Kozae

A.M.

and El-Bably

M.K.

, Generalized covering approximation space and near concepts with some applications, Journal of Applied Computing and Informatics 12(1) (2016), 51–69.

41.

Skowron

, On the topology in information systems, Bull Polish Academy Sciences-Mathematics 36 (1988), 477–480.

42.

Wiweger

, On topological rough sets, Bulletin of the Polish Academy of Sciences Mathematics 37 (1988), 51–62.

43.

Lin

T.Y.

, Topological and fuzzy rough sets in: Slowinski R (ed) Decision Support by Experience Application of the Rough Sets Theory. Kluwer, Norwell, (1992), 287–304.

44.

Polkowski

, On fractals in information systems: The first step In: Terano T., Ohsawa Y., Nishida T., Namatame A., Tsumoto S., Washio T. (eds) New Frontiers in Artificial Intelligence. JSAI 2001, Lecture Notes in Computer Science, Vol. 2253, Springer, Berlin, Heidelberg.

45.

Polkowski

and Semeniuk-Polkowska

, Some remarks on sets of communicating sequential processes in topological rough set framework, Fundamenta Informaticae 60 (2004), 291–305.

46.

Yang

X.B.

, Song

X.N.

and Dou

H.L.

, Multi-granulation rough set: from crisp to fuzzy case, Annals of Fuzzy Mathematics and Informatics 1(1) (2011), 55–70.

47.

Kortelainen

, On relationship between modified sets, topology spaces and rough sets, Fuzzy Sets and Systems 61(1) (1994), 91–95.

48.

Jarvinen

and Kortelainen

, A unifying study between model-like operators, topologies, and fuzzy sets, Fuzzy Sets and Systems 158(11) (2007), 1217–1225.

49.

Thivagar

M.L.

and Richard

, On nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention 1(1) (2013), 31–37.

50.

, On some types of neighborhood-related covering rough sets, International Journal of Approximate Reasoning 53 (2012), 901–911.

51.

Nawar

A.S.

, El-Bably

M.K.

and El-Atik

A.A.

, Certain types of coverings based rough sets with application, Journal of Intelligent and Fuzzy Systems 39(3) (2020), 3085–3098.

52.

Atef

, Khalil

A.M.

, Li

S.G.

, Azzam

and El Atik

, Comparison of six types of rough approximations based on-neighborhood space and-adhesion neighborhood space, Journal of Intelligent and Fuzzy Systems 39(3) (2020), 4515–4531.

53.

Khalil

A.M.

, Li

, Lin

, et al., A new expert system in prediction of lung cancer disease based on fuzzy soft sets, Soft Computing 24 (2020), 14179–14207.

54.

De Baets

and Kerre

, A revision of Bandler-Kohout compositions of relations, Mathematica Pannonica 4(1) (1993), 59–78.

A topological reduction for predicting of a lung cancer disease based on generalized rough sets

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Pawlak’s rough sets

2.2 Yao’s rough sets

2.3 Allam et al.’s rough sets

3 Generalized rough sets based on adhesion neighborhoods via a binary relation

Table 3.1 Comparison between the proposed approaches and the previous methods (namely, Yao and Allam et al. approaches)

5 Medical applications for diagnosis of lung cancer disease

5.1 Data of medical diagnosis of a lung cancer disease

5.3 Topological reduction of attributes

Footnotes

Acknowledgments

Compliance with ethical standards

Conflict of interest

Ethical approval

Availability of data and material

Funding

References

Table 3.1
Comparison between the proposed approaches and the previous methods (namely, Yao and Allam et al. approaches)