Corrigendum to “Comparison of six types of rough approximations based on j -neighborhood space and j -adhesion neighborhood space"

Abstract

The main aims of this paper are to show that some results presented in [1] are erroneous. To this end, we provide some counterexamples to demonstrate our claim, and give the correct form of the incorrect results in [1]. Also, some improvements for the definition of accuracy measure is proposed. Furthermore, we show that the relationships given in the three figures need not be true in general, and determine the conditions under which they are correct. Finally, a medical application in the decision-making of the diagnosis of dengue fever is examined.

Keywords

Rough sets lower and upper approximations accuracy measure

1 Introduction and preliminaries

The concept of rough sets was proposed by Pawlak ([2, 3]) to cope with the uncertainty of data in the information system. The rationale of this concept is the lower and upper approximations which are defined using the equivalence relation. Then, many types of lower and upper approximations were initiated by replacing the equivalence relation with some types of binary relations. Yao [4] introduced the concepts of right and left neighborhoods which are called sometimes after and fore sets. Later on, Allam et al. defined the concepts of minimal right neighborhoods and minimal left neighborhoods in [5] and [6], respectively.

In 2014, Abd El-Monsef et al. [7] presented the concept of j-neighborhood space based on different types of neighborhoods induced from a binary relation. They also constructed new generalized rough sets based on different topologies induced from a binary relation. In 2017, Amer et al. [8] exploited some topological notions called "near open sets" to define j-near approximations in a j-neighborhood space. The concept of adhesion set was introduced in covering-based rough set in [9] for the first time. Nawar et al. [10] extended this notion to any binary relation and then introduced certain types of covering-based rough sets based on generalized covering approximation space [11]. Atef et al. [1] proposed and studied new types of neighborhoods, namely j-adhesion neighborhoods. Based on these neighborhoods, they suggested six different definitions for generalized rough approximations and studied their properties. Al-shami [12] introduced the concept of C_j-neighborhoods and applied to initiate new rough set models. Al-shami et al. [14] presented and studied new types of neighborhoods, namely E_j-neighborhoods. Hosny et al. [16] generated some topologies using the concepts of E_j-neighbourhoods and ideals. Salama et al. [34] S-n investigated topological approaches for rough continuous functions and their applications. Al-shami et al. [13] studied the approximations and accuracy measures induced from N_j-neighborhoods and E_j-neighborhoods. Recently, El-Bably and Al-shami [15] have introduced some kinds of generalized rough sets based on neighborhoods and presented some medical applications.

However, we note some errors in [1]. The first type of errors appeared in the properties of new lower and upper approximations such as property (5) of Proposition 3.15, the property (10) in Proposition 4.7 and Proposition 4.9 and the properties (8) and (10) in Proposition 4.15 and Proposition 4.17.

The second type of errors emerged in the accuracy of some subsets which are given from the lower and upper approximations suggested in Definition 3.1. Sometimes this accuracy is greater than one which represents a contradiction with the meaning of the accuracy of a subset according to Pawlak’s assumptions [2]. We remove this shortcoming by redefining the previous lower and upper approximations. With the help of a concrete example, we show that a property (5) of Proposition 3.15 need not true in general, and then we prove the correct form of this property.

The third type of errors existed in Section 4, we prove the equivalence among Definition 4.3, Definition 4.11, and Definition 4.19 in the cases of j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}. Consequently, we obtain the equivalence among Definition 4.5, Definition 4.13, and Definition 4.21 in these cases of j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}. Besides, we proved some new properties of j-adhesion neighborhood (resp. j-adhesion approximations). Moreover, we found some errors and contradictions in Figure 1, Figure 2, and Figure 3 of [1]. Therefore, we explain under what conditions the relationships studied in these figures are true.

Multi-attribute decision-making (MADM) is a crucial topic in decision-making theory. MADM’s main purpose is to assess the performance of different options in multi-attribute environments [17]. To create an evaluation matrix, a decision maker (DM) evaluates each alternative based on a set of attributes. Many decision-making models have been proposed to help a DM reach a reasonable decision. These models are typically built on traditional 2WD platforms. Yao ([18 –20]) recently has proposed a new theory called 3WD, which is based on thinking in threes. By introducing a third delayed decision option, 3WD can effectively reduce decision risks. The emergence of 3WD opens up a new path and provides new opportunities for MADM study. After then, Zhan et al. [21] succeeded in incorporate 3WD into multi-attribute decision-making (MADM) based on an outranking relation. Besides, Zhan et al. [22], in 2020, proposed three strategies to design a new 3WD model for MADM. After then, an investigation on Wu-Leung multi-scale information systems and multi-expert group decision-making was discussed by Zhan et al. [23]. Moreover, Zhan et al. [24] suggested a novel three-way decision model based on utility theory in incomplete fuzzy decision systems. A novel type of soft rough covering and its application to multicriteria group decision-making was investigated by J. Zhan, J. Carlos and R. Alcantud [25]. Malware analysis aims to identify malware by examining applications behavior on the host operating system. So, Nauman et al. [26] used probabilistic rough sets to define a three-way decision making approach to malware analysis. Ye et al. [27] presented a novel 3WD method to solve MCDM problems in fuzzy information systems. On multi-criteria decision-making and its applications can found in ([28, 29]).

Accordingly, we present a simple medical application of decision making about the diagnosis of dengue fever. We use the introduced approximations in the current paper to be accurate tools for removing the vagueness of the data. In fact, we proved that these approaches are more accurate and stronger than the other methods. Therefore, we can say that these approximations may be useful tools in MADM.

Definition 1.1. [17] Let R be a binary relation on a non-empty finite set Ω. The j-neighborhoods of x ∈ Ω (denoted by N_j (x)) are defined for each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈} as follows:

$N_{j_{1}} (x) = {y \in Ω : x R y}$ .

$N_{j_{2}} (x) = {y \in Ω : y R x}$ .

$N_{j_{3}} (x) = ⋂_{x \in N_{j_{1}} (y)} N_{j_{1}} (y)$ .

$N_{j_{4}} (x) = ⋂_{x \in N_{j_{2}} (y)} N_{j_{2}} (y)$ .

$N_{j_{5}} (x) = N_{j_{1}} (x) ⋃ N_{j_{2}} (x)$ .

$N_{j_{6}} (x) = N_{j_{1}} (x) ⋂ N_{j_{2}} (x)$ .

$N_{j_{7}} (x) = N_{j_{3}} (x) ⋃ N_{j_{4}} (x)$ .

$N_{j_{8}} (x) = N_{j_{3}} (x) ⋂ N_{j_{4}} (x)$ .

Definition 1.2. [7] Let R be a binary relation on a non-empty finite set Ω and ζ_j : Ω ⟶ 2^Ω be a mapping which assigns for each x ∈ Ω it’s j-neighborhood in 2^Ω. The triple (Ω, R, ζ_j) is called a j-neighborhood space (briefly, j-NS)

Definition 1.3. [1] Let R be a binary relation on a non-empty finite set Ω. The j-adhesion neighborhoods of x ∈ Ω (denoted by P_j (x)) are defined for each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈} as follows:

$P_{j_{1}} (x) = {y \in Ω : N_{j_{1}} (y) = N_{j_{1}} (x)}$ .

$P_{j_{2}} (x) = {y \in Ω : N_{j_{2}} (y) = N_{j_{2}} (x)}$ .

$P_{j_{3}} (x) = {y \in Ω : ⋂_{x \in N_{j_{1}} (y)} N_{j_{1}} (y) = ⋂_{y \in N_{j_{1}} (x)} N_{j_{1}} (x)}$ .

$P_{j_{4}} (x) = {y \in Ω : ⋂_{x \in N_{j_{2}} (y)} N_{j_{2}} (y) = ⋂_{y \in N_{j_{2}} (x)} N_{j_{2}} (x)}$ .

$P_{j_{5}} (x) = P_{j_{1}} (x) ⋃ P_{j_{2}} (x)$ .

$P_{j_{6}} (x) = P_{j_{1}} (x) ⋂ P_{j_{2}} (x)$ .

$P_{j_{7}} (x) = P_{j_{3}} (x) ⋃ P_{j_{4}} (x)$ .

$P_{j_{8}} (x) = P_{j_{3}} (x) ⋂ P_{j_{4}} (x)$ .

The following two definitions was given in [1] as Definition 3.1 and Definition 3.4.

Definition 1.4. Let (Ω, R, ζ_j) be a j-neighborhood space. A type 1 of rough set based on j-neighborhood (briefly, type 1 j-neighborhood rough set) of X ⊆ Ω in (Ω, R, ζ_j) or with respect to ζ_j is a pair $(^{1} N_{jR} (X),^{1} N_{j}^{R} (X))$ , defined by

$^{1} N_{jR} (X) = {x \in Ω : N_{j} (x) \subseteq X}$ (called the type 1 lower approximation of X) and

$^{1} N_{j}^{R} (X) = {x \in Ω : N_{j} (x) ⋂ X \neq \emptyset}$ (called the type 1 upper approximation of X)

Definition 1.5. Let (Ω, R, ζ_j) be a j-neighborhood space. A type 1 j-neighborhood boundary region (briefly, $^{1} B_{j}$ ) and a type 1 j-neighborhood accuracy measure (briefly, $^{1} A_{j}$ ) of X ⊆ Ω defined by

$^{1} B_{j} =$ $^{1} N_{j}^{R} (X) -$ $^{1} N_{jR} (X)$ .

$^{1} A_{j} (X) = \frac{∣^{1} N_{jR} (X) ∣}{∣^{1} N_{j}^{R} (X) ∣}$ .

Proposition 1.6. [2, 3] Let R be an equivalence relation on a non-empty finite set Ω and X, Y ⊆ Ω. Then the following properties hold.

$\underline{R} (X) \subseteq X$

$X \subseteq \bar{R} (X)$

$\underline{R} (\emptyset) = \emptyset$

$\bar{R} (\emptyset) = \emptyset$

$\underline{R} (Ω) = Ω$

$\bar{R} (Ω) = Ω$

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

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

$\underline{R} (X \cap Y) = \underline{R} (X) \cap \underline{R} (Y)$

$\bar{R} (X \cap Y) \subseteq \bar{R} (X) \cap \bar{R} (Y)$

$\underline{R} (X) \cup \underline{R} (Y) \subseteq \underline{R} (X \cup Y)$

$\bar{R} (X \cup Y) = \bar{R} (X) \cup \bar{R} (Y)$

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

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

$\underline{R} (\underline{R} (X)) = \underline{R} (X)$

$\bar{R} (\bar{R} (X)) = \bar{R} (X)$

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

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

$\forall K \in Ω / R \Rightarrow \underline{R} (K) = K$

$\forall K \in Ω / R \Rightarrow \bar{R} (K) = K$

2 Corrections for some results

We devote this section to correct some errors of [1] and remove the defect appears when calculating the accuracy value. We provide some illustrative examples to clarify our notes.

It is well known that the accuracy of a subset in rough set theory lies in the closed interval [0, 1] as in [2, 3]. In the literature, there are many formulations of calculating the accuracy of a subset; all of them requires that the accuracy be greater or equal to zero and less or equal to one. However, we note that the accuracy value according to Definition 3.4 of [1] may be greater than one, and this is a contradicts the meaning of the accuracy of a subset. The following example illustrates this matter.

Example 2.1. In Example 3.2 of [6], the authors calculated that $^{1} N_{j 8 R} (X) = {a, b, d}$ and $^{1} N_{j 8}^{R} (X) = {a, b}$ , where X = {a, b}. In this case, the accuracy of X according to Definition 3.4: $^{1} A_{j 8} (X) = \frac{∣^{1} N_{j 8 R} (X) ∣}{∣^{1} N_{j 8}^{R} (X) ∣} = \frac{3}{2} > 1$ .

In fact this shortcoming follows from Definition 3.1; therefore, we restate Definition 3.1 as follows.

Definition 2.2. Let (Ω, R, ζ_j) be a j-neighborhood space. A type 1 of rough set based on j-neighborhood (briefly, type 1 j-neighborhood rough set) of X ⊆ Ω in (Ω, R, ζ_j) or with respect to ζ_j is a pair $(^{1} N_{jR} (X)$ , $^{1} N_{j}^{R} (X))$ , defined by

$^{1} N_{jR} (X) = X ⋂ {x \in Ω : N_{j} (x) \subseteq X}$ (called the type 1 lower approximation of X) and

$^{1} N_{j}^{R} (X) = X ⋃ {x \in Ω : N_{j} (x) ⋂ X \neq \emptyset}$ (called the type 1 upper approximation of X)

According to Definition (2), it can be seen that the accuracy of any subset of Ω must be in the closed interval [0, 1].

We note that item (5) in Proposition 3.15 of [1] need not true in general as the following example shows.

Example 2.3. Let R = {(a, b), (a, d), (a, e), (b, c), (b, d), (c, c), (c, e), (d, a), (d, b), (d, c), (e, d)} is a binary relation on Ω = {a, b, c, d, e}. Take j = j₁. Then we obtain

$N_{j_{1}} (a) = {b, d, e}$ ;

$N_{j_{1}} (b) = {c, d}$

$N_{j_{1}} (c) = {c, e}$

$N_{j_{1}} (d) = {a, b, c}$

$N_{j_{1}} (e) = {d}$

Now, let X₁ = {a} and X₂ = {e}. Then $^{2} N_{j_{2}}^{R} (X_{1}) = {a}$ and $^{2} N_{j_{2}}^{R} (X_{2}) = {e}$ . But $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2}) = {a, b, e}$ . Therefore, $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2}) \subseteq$ $^{2} N_{j_{2}}^{R} (X_{1}) ⋃$ $^{2} N_{j_{2}}^{R} (X_{2})$ .

The following result gives the correct form of item (5) in Proposition 3.15 of [6].

Proposition 2.4. Let (Ω, R, ζ_j) be a j-neighborhood space. Then $^{2} N_{j_{2}}^{R} (X_{1}) ⋃$ $^{2} N_{j_{2}}^{R} (X_{2}) \subseteq$ $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2})$ for each X₁, X₂ ⊆ Ω.

Proof. Since $^{2} N_{j_{2}}^{R} (X_{1}) \subseteq$ $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2})$ and $^{2} N_{j_{2}}^{R} (X_{2}) \subseteq$ $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2})$ , then $^{2} N_{j_{2}}^{R} (X_{1}) ⋃$ $^{2} N_{j_{2}}^{R} (X_{2}) \subseteq$ $^{2} N_{j_{2}}^{R} (X_{1} ⋃ X_{2})$ , as required.□

We construct the following example to show that the property (10) in Proposition 4.7 and Proposition 4.9 and the properties (8) and (10) in Proposition 4.15 and Proposition 4.17 need not be true in general.

Example 2.5. Let R = {(c, a), (d, a), (c, b), (d, c), (c, d), (d, d)} be a binary relation on Ω = {a, b, c, d}.

Table 2
The j-adhesion neighborhoods of x ∈ Ω\label 2

x ∈ Ω a b c d

$P_{j_{1}} (x)$ {a,b} {a,b} {c} {d}

$P_{j_{2}} (x)$ {a,d} {b} {c} {a, d}

$P_{j_{3}} (x)$ {a,d} {b} {c} {a, d}

$P_{j_{4}} (x)$ {a,b} {a,b} {c} {d}

$P_{j_{5}} (x)$ {a,b,d} {a,b} {c} {a, d}

$P_{j_{6}} (x)$ {a} {b} {c} {d}

$P_{j_{7}} (x)$ {a,b,d} {a,b} {c} {a, d}

$P_{j_{8}} (x)$ {a} {b} {c} {d}

x ∈ Ω	a	b	c	d
$P_{j_{1}} (x)$	{a,b}	{a,b}	{c}	{d}
$P_{j_{2}} (x)$	{a,d}	{b}	{c}	{a, d}
$P_{j_{3}} (x)$	{a,d}	{b}	{c}	{a, d}
$P_{j_{4}} (x)$	{a,b}	{a,b}	{c}	{d}
$P_{j_{5}} (x)$	{a,b,d}	{a,b}	{c}	{a, d}
$P_{j_{6}} (x)$	{a}	{b}	{c}	{d}
$P_{j_{7}} (x)$	{a,b,d}	{a,b}	{c}	{a, d}
$P_{j_{8}} (x)$	{a}	{b}	{c}	{d}

Let X = {a, b, c}. We calculate for each j ∈ {j₅, j₇} the following:

$^{4} N_{jR} (X) = {b, c}$ , and $^{4} N_{j}^{R} (^{4} N_{jR} (X)) = {a, b, c} = X$ .

$^{5} N_{jR} (X) = {a, b, c} = X$ , and $^{5} N_{j}^{R} (^{5} N_{jR} (X)) = Ω$ .

Therefore, we have the following:

$^{4} N_{j}^{R} (^{4} N_{jR} (X)) \neq$ $^{4} N_{jR} (X)$ .

$^{5} N_{j}^{R} (^{5} N_{jR} (X)) \neq$ $^{5} N_{jR} (X)$ .

$^{5} N_{j}^{R} (^{5} N_{jR} (X)) \subseteq X$ .

By duality of the approximations, we have the following

$^{4} N_{jR} (^{4} N_{j}^{R} (X)) \neq$ $^{4} N_{j}^{R} (X)$ .

$^{5} N_{jR} (^{5} N_{j}^{R} (X)) \neq$ $^{5} N_{j}^{R} (X)$ .

X⊆ $^{5} N_{jR} (^{5} N_{j}^{R} (X))$ .

Also, the property (9) of Proposition 4.7 and Proposition 4.9 need not be true in general. To show that, let A = {a, b} ⊆ Ω. Then for each j ∈ {j₅, j₇}, we have $^{4} N_{jR} (A) = {b}$ , and $^{4} N_{jR} (^{4} N_{jR} (A)) = \emptyset$ . Therefore, $^{4} N_{jR} (A) \neq$ $^{4} N_{jR} (^{4} N_{jR} (A))$ . By duality of the approximations, we also have $^{4} N_{j}^{R} (A) \neq$ $^{4} N_{j}^{R} (^{4} N_{j}^{R} (A))$ .

3 Further properties of j-adhesion neighborhoods

In what follows, we prove that Definition 4.3, Definition 4.11, and Definition 4.19 in [1] are equivalent in the cases of j = j₁, j₂, j₃, j₄, j₆, j₈. This leads to that Definition 4.5, Definition 4.13, and Definition 4.21 are also equivalent in the cases of j = j₁, j₂, j₃, j₄, j₆, j₈. In addition, we amend the relationships given Figs. 1-3.

Before that, we need the following auxiliary results.

Theorem 3.1. Let (Ω, R, ζ_j) be a j-neighborhood space and x, y ∈ Ω. Then for each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈}, we have the following results.

$x \in P_{j} (x)$ for each x ∈ Ω.

$y \in P_{j} (x)$ iff P_j (x) = P_j (y) for each j ≠ j₅ and j ≠ j₇.

Proof.

Straightforward.

Necessity: let $y \in P_{j} (x)$ . Then $N_{j} (y) = N_{j} (x)$ (1) Let $z \in P_{j} (y)$ . Then $N_{j} (z) = N_{j} (y)$ . It follows from 1 that $N_{j} (z) = N_{j} (x)$ . This means that $z \in P_{j} (x)$ . Thus, P_j (y) ⊆ P_j (x). Similarly, one can prove that P_j (x) ⊆ P_j (y). Hence, the necessary part is proved. The Sufficient part is obvious.□

Corollary 3.2. The class $σ (Ω) = {P_{j} (x) : x \in Ω}$ forms a partition of Ω for each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈}.

In Example 2.5, note that $a \in P_{j_{5}} (b)$ , but $P_{j_{5}} (a) \neq P_{j_{5}} (b)$ . Also, $a \in P_{j_{7}} (b)$ , but $P_{j_{7}} (a) \neq P_{j_{7}} (b)$ . Moreover, σ (Ω) = {{c} , {a, b} , {a, d} , {a, b, d}} is not a partition of Ω. This shows that the converse item (ii) of Theorem 3.1 and Corollary 3.2 is not true in case of j ∈ {j₅, j₇}.

Now, we are in a position to prove the following result which demonstrates the equivalence among Definition 4.3, Definition 4.11, and Definition 4.19 in the cases of j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}.

Theorem 3.3. Let (Ω, R, ζ_j) be a j-neighborhood space and X ⊆ Ω. Then for each j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}, we have the following results.

$^{4} N_{jR} (X) =$ $^{5} N_{jR} (X) =$ $^{6} N_{jR} (X)$

$^{4} N_{j}^{R} (X) =$ $^{5} N_{j}^{R} (X) =$ $^{6} N_{j}^{R} (X)$

Proof. We only prove (i). The other one can be made similarly.

$^{4} N_{jR} (X) \subseteq$ $^{5} N_{jR} (X)$ : Let x∈ $^{4} N_{jR} (X)$ . Then x ∈ Ω such that $P_{j} (x) \subseteq X$ . Therefore, $x \in ⋃ {P_{j} (y) : P_{j} (y) \subseteq X} =$ $^{5} N_{jR} (X)$ . Thus $^{4} N_{jR} (X) \subseteq^{5} N_{jR} (X)$ (2)

$^{5} N_{jR} (X) \subseteq$ $^{6} N_{jR} (X)$ : Let x∈ $^{5} N_{jR} (X)$ . Then $x \in ⋃ {P_{j} (y) : P_{jR} (y) \subseteq X}$ . This means that x ∈ X. It follows from Theorem 3.1 that $x \in P_{j} (x)$ . It follows from Corollary 3.2 that $x \in P_{j} (x)$ and $x \notin P_{j} (y)$ for each y ≠ x. Therefore, $P_{j} (x) \subseteq X$ . Thus, $P_{j} (x) ⋂ X^{c} = \emptyset$ . This implies that $x \in [⋃ {P_{j} (y) : P_{j} (y) ⋂ X^{c} \neq \emptyset}]^{c} =$ $^{6} N_{jR} (X)$ . Hence, $^{5} N_{jR} (X) \subseteq^{6} {NR}_{j} (X)$ (3)

$^{6} N_{jR} (X) \subseteq$ $^{4} N_{jR} (X)$ : Let x∈ $^{6} N_{jR} (X)$ . Then $x \in [⋃ {P_{j} (y) : P_{j} (y) ⋂ X^{c} = \emptyset}]^{c}$ . This implies that $P_{j} (x) ⋂ X^{c} \neq \emptyset$ . Therefore, $P_{j} (x) \subseteq X$ . Thus, x∈ $^{4} N_{jR} (X)$ . Hence, we obtain $^{6} N_{jR} (X) \subseteq^{4} N_{jR} (X)$ (4)

From 2, 3 and 4, we obtain the desired result.□

Remark 3.4. To show that the above theorem is not true in case of j ∈ {j₅, j₇}, let X = {a, b, c} and Y = {c, d}. Then, it can be seen from Example 2.5 that $^{4} N_{j_{5} R} (X) = {b, c}$ , $^{5} N_{j_{5} R} (X) = {a, b, c} = X$ , and $^{6} N_{j_{5} R} (X) = {c}$ . Also, $^{4} N_{j_{5}}^{R} (Y) = {a, c, d}$ , $^{5} N_{j_{5}}^{R} (Y) = {c, d}$ , and $^{6} N_{j_{5}}^{R} (Y) = Ω$ .

In the following, we illustrate that the definition of j-adhesion approximations gives only four different rough set approximations for each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈}. In other words, different eight j-adhesion approximations are reduced into four different rough set approximations.

Theorem 3.5. Let (Ω, R, ζ_j) be a j-neighborhood space and x ∈ Ω. Then we have the following results.

$P_{j_{2}} (x) = P_{j_{3}} (x)$ .

$P_{j_{1}} (x) = P_{j_{4}} (x)$ .

Proof. We only prove (i). The other one can be made similarly.

Let $y \in P_{j_{3}} (x)$ . Then $P_{j_{3}} (x) = P_{j_{3}} (y)$ and $N_{j_{3}} (x) = N_{j_{3}} (y)$ . Now, we will prove that $P_{j_{2}} (x) = P_{j_{2}} (y)$ . Suppose that $P_{j_{2}} (x) \neq P_{j_{2}} (y)$ . Then $y \notin P_{j_{2}} (x)$ and $x \notin P_{j_{2}} (y)$ . Therefore, $N_{j_{2}} (x) \neq N_{j_{2}} (y)$ . This means that there exists $z \in N_{j_{2}} (y)$ such that $z \notin N_{j_{2}} (x)$ . Thus, $y \in N_{j_{2}} (z)$ and $x \notin N_{j_{2}} (z)$ . Consequently, $z \in N_{j_{3}} (y)$ and $z \notin N_{j_{3}} (x)$ . This leads to $N_{j_{3}} (x) \neq N_{j_{3}} (y)$ . But this is contradicts assumption. Hence, $y \in P_{j_{2}} (x)$ and this proves that $P_{j_{3}} (x) \subseteq P_{j_{2}} (x)$ , as required. Similarly, one can prove that $P_{j_{2}} (x) \subseteq P_{j_{3}} (x)$ .□

Corollary 3.6. Let (Ω, R, ζ_j) be a j-neighborhood space and x ∈ Ω. Then we have the following results.

$P_{j_{5}} (x) = P_{j_{7}} (x)$ .

$P_{j_{6}} (x) = P_{j_{8}} (x)$ .

Proposition 3.7. Let (Ω, R, ζ_j) be a j-neighborhood space. Then for each X ⊆ Ω and n ∈ {4, 5, 6}, we have the following results.

$^{n} N_{j_{1} R} (X) =$ $^{n} N_{j_{4} R} (X)$ .

$^{n} N_{j_{2} R} (X) =$ $^{n} N_{j_{3} R} (X)$ .

$^{n} N_{j_{5} R} (X) =$ $^{n} N_{j_{7} R} (X)$ .

$^{n} N_{j_{6} R} (X) =$ $^{n} N_{j_{8} R} (X)$ .

$^{n} N_{j_{1}}^{R} (X) =$ $^{n} N_{j_{4}}^{R} (X)$ .

$^{n} N_{j_{2}}^{R} (X) =$ $^{n} N_{j_{3}}^{R} (X)$ .

$^{n} N_{j_{5}}^{R} (X) =$ $^{n} N_{j_{7}}^{R} (X)$ .

$^{n} N_{j_{6}}^{R} (X) =$ $^{n} N_{j_{8}}^{R} (X)$ .

Proof. The proof follows from Theorem 3.5 and Corollary 3.6.□ The following example shows that the relationships given in Figs. 1-3 of [1] need not be true in general.

Example 3.8. Let R be the same as given in Example 2.5. Then we note that $N_{j}$ and $P_{j}$ are independent of each other.

Now, we explain under what conditions the relationships given in Figs. 1-3 of [6] are correct.

Lemma 3.9. Let (Ω, R, ζ_j) be a j-neighborhood space such that R is reflexive. Then $P_{j} (x) \subseteq N_{j} (x)$ for each x ∈ Ω and each j ∈ {j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈}.

Proof. We will prove the lemma in the case of j = j₁. The other cases follow similarly.

Let $y \in P_{j_{1}} (x)$ , then $N_{j_{1}} (y) = N_{j_{1}} (x)$ . The reflexivity of R implies that $y \in N_{j_{1}} (y)$ . Therefore, $y \in N_{j_{1}} (x)$ . Thus, $P_{j} (x) \subseteq N_{j} (x)$ , as required.□

The proofs of the following two results follow from Theorem 3.3 and Lemma 3.9.

Proposition 3.10. Let (Ω, R, ζ_j) be a j-neighborhood space such that R is reflexive. Then we have the following properties for each X ⊆ Ω and each j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}.

$^{1} N_{jR} (X) \subseteq$ $^{4} N_{jR} (X) \subseteq X \subseteq$ $^{4} N_{j}^{R} (X) \subseteq$ $^{1} N_{j}^{R} (X)$ .

$^{2} N_{jR} (X) \subseteq$ $^{4} N_{jR} (X) \subseteq X \subseteq$ $^{4} N_{j}^{R} (X) \subseteq$ $^{2} N_{j}^{R} (X)$ .

$^{3} N_{jR} (X) \subseteq$ $^{4} N_{jR} (X) \subseteq X \subseteq$ $^{4} N_{j}^{R} (X) \subseteq$ $^{3} N_{j}^{R} (X)$ .

Proposition 3.11. Let (Ω, R, ζ_j) be a j-neighborhood space such that R is reflexive. Then we have the following properties for each X ⊆ Ω and each j ∈ {j₅, j₇}.

$^{1} N_{jR} (X) \subseteq$ $^{4} N_{jR} (X) \subseteq X \subseteq$ $^{4} N_{j}^{R} (X) \subseteq$ $^{1} N_{j}^{R} (X)$ .

$^{2} N_{jR} (X) \subseteq$ $^{5} N_{jR} (X) \subseteq X \subseteq$ $^{5} N_{j}^{R} (X) \subseteq$ $^{2} N_{j}^{R} (X)$ .

$^{3} N_{jR} (X) \subseteq$ $^{6} N_{jR} (X) \subseteq X \subseteq$ $^{6} N_{j}^{R} (X) \subseteq$ $^{3} N_{j}^{R} (X)$ .

4 Medical application to dengue fever disease

In this section, we are considering the problem of dengue fever disease which is transmitted to humans via virus-carrying dengue mosquitoes ([30, 31]). The data given in Table 3 is for eight patients of dengue fever P = {m₁, m₂, m₃, m₄, m₅, m₆, m₇, m₈}. The set of attributes (symptoms of dengue fever) is A = {J, F, S, H}, where J, F, S and H respectively denote muscle and joint pains, fever, characteristic skin rash, and headache. The attributes have two values: ‘✓’ and squo × ′ which respectively mean that the patient has symptoms and the patient has no symptoms. The made decision also has the same two values with the meaning of possessing dengue fever disease or not.

Table 3
Dengue fever information system [32]

P J F S H Dengue fever

m ₁ ✓ ✓ ✓ × ✓

m ₂ ✓ × × × ×

m ₃ ✓ × × × ✓

m ₄ × × × ✓ ×

m ₅ × ✓ ✓ × ×

m ₆ ✓ ✓ × ✓ ✓

m ₇ ✓ ✓ × × ×

m ₈ ✓ ✓ × ✓ ✓

P	J	F	S	H	Dengue fever
m ₁	✓	✓	✓	×	✓
m ₂	✓	×	×	×	×
m ₃	✓	×	×	×	✓
m ₄	×	×	×	✓	×
m ₅	×	✓	✓	×	×
m ₆	✓	✓	×	✓	✓
m ₇	✓	✓	×	×	×
m ₈	✓	✓	×	✓	✓

We determine the symptoms of every patient by a map v : P → 2^A such that a symptom belongs to v (m_i) if the patient m_i has this symptom.

From Table 3, we obtain the symptoms of every patient as follows:

v (m₁) = {J, F, S}, v (m₂) = v (m₃) = {J}, v (m₄) = {H}, v (m₅) = {F, S}, v (m₆) = v (m₈) = {J, F, H}, and v (m₇) = {J, F}.

Now, we define a binary relation between the patients depending on the map v as follows. $m_{x} {Rm}_{y} \Leftrightarrow v (m_{x}) \subseteq v (m_{y}) .$ (5)

Note that: The relation is identified according to the viewpoint of system’s experts.

According to the relation given in 5, we obtain R = {(m₁, m₁), (m₂, m₂), (m₂, m₁), (m₂, m₃), (m₂, m₆), (m₂, m₇), (m₂, m₈), (m₃, m₃), (m₃, m₁), (m₃, m₂), (m₃, m₆), (m₃, m₇), (m₃, m₈), (m₄, m₄), (m₄, m₆), (m₄, m₈), (m₅, m₅), (m₅, m₁), (m₆, m₆), (m₆, m₈), (m₇, m₇), (m₇, m₁), (m₇, m₆), (m₇, m₈), (m₈, m₈), (m₈, m₆)}.

Therefore, we will compare the suggested method $(^{4} N_{j_{1} R} (X)$ , $^{4} N_{j_{1}}^{R} (X))$ given in Definitions 4.3 and 4.11 in [1], and we compare it with Yao’s method [4] and Dai et al.’s methods [33]:

First, let us recall the basic definitions of Yao and Dai et al.’s methods:

Definition 4.1. Let Ω be a finite set and R be a binary relation on Ω. Then the lower and upper approximations, boundary region and accuracy measure of a subset X are given respectively by

$R_{*} (X) = {x \in Ω : N_{j_{1}} (x) \subseteq X}$ ,

$R^{*} (X) = {x \in Ω : N_{j_{1}} (x) \cap X \neq \emptyset},$

$B_{*} (X) =$ R_∗ (X), and $A_{*} (X) = \frac{∣ R_{*} (X) ∣}{∣ R^{*} (X) ∣}$

Definition 4.2. [33] Let Ω be a finite set and R be a binary relation on Ω. Then the maximal neighborhood of an element x ∈ X is given by $N_{m} (x) = {\begin{matrix} ⋃_{x \in N_{j_{1}} (y)} N_{j_{1}} (y) & if \exists y such that x \in N_{j_{1}} (y) \\ \emptyset & Otherwise . \end{matrix}$

Definition 4.3. [33] Let Ω be a finite set and R be a binary relation on Ω. Then the maximal- lower and maximal-upper approximations, maximal-boundary region and maximal-accuracy measure of a subset X are given respectively by

$R_{m} (X) = {x \in Ω : N_{m} (x) \subseteq X}$ ,

$R^{m} (X) = {x \in Ω : N_{m} (x) \cap X \neq \emptyset},$

$B_{m} (X) =$ , and $A_{m} (X) = \frac{∣ R_{m} (X) ∣}{∣ R_{m} (X) ∣}$

In the following, we calculate the j₁-neighborhoods $N_{j_{1}}$ , maximal neighborhoods and j₁-adhesion neighborhoods $P_{j_{1}}$ of all elements in P. $N_{j_{1}} (m_{1}) = {m_{1}}$ , $N_{j_{1}} (m_{2})$ $= N_{j_{1}} (m_{3}) = {m_{1}$ , m₂, m₃, m₆, m₇, m₈}, $N_{j_{1}} (m_{4}) = {m_{4}$ , m₆, m₈}, $N_{j_{1}} (m_{5}) = {m_{1}$ , m₅}, $N_{j_{1}} (m_{6}) = N_{j_{1}} (m_{8}) = {m_{6}$ , m₈} and $N_{j_{1}} (m_{7}) = {m_{1}$ , m₆, m₇, m₈}.

$N_{m} (m_{1}) = {m_{1}$ , m₂, m₃, m₅, m₆, m₇, m₈}, $N_{m} (m_{2})$ $= N_{m} (m_{3}) = N_{m} (m_{7}) = N_{m} (m_{3}) = N_{m} (m_{8}) = {m_{1}$ , m₂, m₃, m₆, m₇, m₈}, $N_{j_{1}} (m_{4}) = {m_{4}$ , m₆, m₈}and $N_{j_{1}} (m_{5}) = {m_{1}$ , m₅}.

$P_{j_{1}} (m_{1}) = {m_{1}}$ , $P_{j_{1}} (m_{2}) = P_{j_{1}} (m_{3}) = {m_{2}$ , m₃}, $P_{j_{1}} (m_{4}) = {m_{4}}$ , $P_{j_{1}} (m_{5}) = {m_{5}}$ , $P_{j_{1}} (m_{6}) = P_{j_{1}} (m_{8}) = {m_{6}$ , m₈} and $P_{j_{1}} (m_{7}) = {m_{7}}$ .

Now, we calculate the approximations, boundary regions and the accuracy measures for A₁ = {m₁, m₃, m₆, m₈} (a set of patients infected with dengue fever) and A₂ = {m₂, m₄ m₅, m₇} (a set of patients infected without dengue fever) by using the suggested method, and we compare it by Yao and Dai et al.’s techniques.

From Table 1, we have two cases are:

Table 1

The j-neighborhoods of x ∈ Ω

x ∈ Ω	a	b	c	d
$N_{j_{1}} (x)$	∅	∅	{a, b, d}	{a,c,d}
$N_{j_{2}} (x)$	{c, d}	{c}	{d}	{c, d}
$N_{j_{3}} (x)$	{a, d}	{a, b, d}	{a, c, d}	{a, d}
$N_{j_{4}} (x)$	∅	∅	{c}	{d}
$N_{j_{5}} (x)$	{c, d}	{c}	{a, b, d}	{a, c, d}
$N_{j_{6}} (x)$	∅	∅	{d}	{c, d}
$N_{j_{7}} (x)$	{a, d}	{a, b, d}	{a, c, d}	{a, d}
$N_{j_{8}} (x)$	∅	∅	{c}	{ d}

Case 1: The set of patients infected with dengue fever A₁ = {m₁, m₃, m₆, m₈}.

Yao’s approach [4]:

By calculating, we obtain R_∗ (A₁) = {m₁, m₆, m₈} and . This means that the boundary region is $B^{*} (A_{1}) = {m_{2}$ , m₃, m₄, m₅, m₇} and the accuracy measure is $A^{*} (A_{1}) = \frac{3}{8}$ .

Dai et al.’s approach [33]:

By calculating, we obtain R_m (A₁) =∅ and R^m (A₁) = P. This means that the boundary region is $B_{m} (A_{1}) = P$ and the accuracy measure is $A_{m} (A_{1}) = 0$ which means that A₁ is totally rough set.

Adhesion approach:

By calculating, we obtain $^{4} N_{j_{1} R} (A_{1}) = {m_{1}$ , m₆, m₈} and $^{4} N_{j_{1}}^{R} (A_{1}) = {m_{1}$ , m₂, m₃, m₆, m₈}. This means that the boundary region is $^{4} B_{j_{1}} (A_{1}) = {m_{2}$ , m₃} and the accuracy measure is $^{4} A_{j_{1}} (A_{1}) = \frac{3}{5}$ .

Case 2: The set of patients don’t infected with dengue fever A₂ = {m₂, m₄, m₅, m₇}.

Yao’s approach:

The approximations of A₂ are R_∗ (A₂) =∅ and R^∗ (A₂) = {m₂, m₃, m₄, m₅, m₇}. This means that the boundary region is $B_{*} (A_{2}) = {m_{2}$ , m₃, m₄, m₅, m₇} and the accuracy measure is $A_{*} (A_{2}) = 0$ .

Dai et al.’s approach:

By calculating, we obtain R_m (A₂) =∅ and R^m (A₂) = P. This means that the boundary region is $B_{m} (A_{1}) = P$ and the accuracy measure is $A_{m} (A_{2}) = 0$ which means that A₂ is totally rough set.

Adhesion approach:

The approximations of A₂ are $^{4} N_{j_{1} R} (A_{1}) = {m_{4}$ , m₅, m₇} and $^{4} N_{j_{1}}^{R} (A_{1}) = {m_{2}$ , m₃, m₄, m₅, m₇}. This means that the boundary region is $^{4} B_{j_{1}} (A_{1}) = {m_{2}$ , m₃} and the accuracy measure is $^{4} A_{j_{1}} (A_{1})$ $= \frac{3}{5}$ .

Observation: From the above comparisons, we notice the following:

Pawlak’s rough set model cannot used in the above application because the used relation is preorder (reflexive and transitive) and Pawlak’s rough set model applied only when the relation is equivalence relation.

There are several approaches which can be used to approximate the sets. According to the obtained results, approximations using j-adhesion neighborhoods is the best one because the boundary region is minimized (or canceled) by maximizing the lower approximation and minimizing the upper approximation. Moreover, the accuracy degree is more accurate than the other types (such as Yao [4] and Dai et al. [33] methods). For example, according to Table 3 (which represents a decision table) the infected patients with dengue fever are {m₁, m₃, m₆, m₈}. Using Dai et al.’s approach we found the boundary region is P which means that all patients in P may be infected with dengue fever; that is, we cannot decide whether the patients are infected with dengue fever or not. Hence, this produces a vagueness in decision making about the medical diagnosis. On the other hand, by using j-adhesion approximations, we get the boundary {m₂, m₃} and hence we minimized the vagueness in the data and also obtain a higher accuracy measure. Accordingly, we can say that the suggested approximations are more accurate than the previous methods for extracting the information and help to eliminate the ambiguity of the data in the real-life problems; especially, in the medical diagnosis which needs accurate decisions.

5 Conclusion

In recent decades, many authors interested in the rough set theory endeavor to reduce the boundary region for the sake of increasing the accuracy measure of decision-making. One of the used techniques is creating new types of neighborhoods and exploiting to establish new types of lower and upper approximations; hence, initiating a new accuracy measure. In this note, we have rectified some alleged properties and improve some results given in [1]. We give some examples to show that some properties of Proposition 3.15, Proposition 4.7, Proposition 4.9, Proposition 4.15 and Proposition 4.17 are not valid in general. Also, we prove some definitions are redundant in the cases of j ∈ {j₁, j₂, j₃, j₄, j₆, j₈}.

In the upcoming works, we will study more properties of j-adhesion neighborhoods and investigate them in topological view. Also, we will define and probe new types of neighborhoods by making use of j-neighborhoods and j-adhesion neighborhoods.

Footnotes

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgment

The authors are grateful to the editor and referees for their valuable comments and suggestions. This publication was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.

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Corrigendum to “Comparison of six types of rough approximations based on j -neighborhood space and j -adhesion neighborhood space"

Abstract

Keywords

1 Introduction and preliminaries

2 Corrections for some results

Table 3 Dengue fever information system [32] P J F S H Dengue fever m 1 ✓ ✓ ✓ × ✓ m 2 ✓ × × × × m 3 ✓ × × × ✓ m 4 × × × ✓ × m 5 × ✓ ✓ × × m 6 ✓ ✓ × ✓ ✓ m 7 ✓ ✓ × × × m 8 ✓ ✓ × ✓ ✓

Footnotes

Conflict of interests

Acknowledgment

References

Table 3
Dengue fever information system [32]

P J F S H Dengue fever

m ₁ ✓ ✓ ✓ × ✓

m ₂ ✓ × × × ×

m ₃ ✓ × × × ✓

m ₄ × × × ✓ ×

m ₅ × ✓ ✓ × ×

m ₆ ✓ ✓ × ✓ ✓

m ₇ ✓ ✓ × × ×

m ₈ ✓ ✓ × ✓ ✓