
There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely I-δβJ-open sets and I-⋀βJ-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.
Citation: Mona Hosny. Rough sets theory via new topological notions based on ideals and applications[J]. AIMS Mathematics, 2022, 7(1): 869-902. doi: 10.3934/math.2022052
[1] | Ashraf S. Nawar, Mostafa A. El-Gayar, Mostafa K. El-Bably, Rodyna A. Hosny . θβ-ideal approximation spaces and their applications. AIMS Mathematics, 2022, 7(2): 2479-2497. doi: 10.3934/math.2022139 |
[2] | Imran Shahzad Khan, Choonkil Park, Abdullah Shoaib, Nasir Shah . A study of fixed point sets based on Z-soft rough covering models. AIMS Mathematics, 2022, 7(7): 13278-13291. doi: 10.3934/math.2022733 |
[3] | Mona Hosny . Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications. AIMS Mathematics, 2022, 7(7): 13104-13138. doi: 10.3934/math.2022724 |
[4] | Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou . New soft rough approximations via ideals and its applications. AIMS Mathematics, 2024, 9(4): 9884-9910. doi: 10.3934/math.2024484 |
[5] | Imran Shahzad Khan, Nasir Shah, Abdullah Shoaib, Poom Kumam, Kanokwan Sitthithakerngkiet . A new approach to the study of fixed points based on soft rough covering graphs. AIMS Mathematics, 2023, 8(9): 20415-20436. doi: 10.3934/math.20231041 |
[6] | R. Mareay, Radwan Abu-Gdairi, M. Badr . Soft rough fuzzy sets based on covering. AIMS Mathematics, 2024, 9(5): 11180-11193. doi: 10.3934/math.2024548 |
[7] | Shoubin Sun, Lingqiang Li, Kai Hu, A. A. Ramadan . L-fuzzy upper approximation operators associated with L-generalized fuzzy remote neighborhood systems of L-fuzzy points. AIMS Mathematics, 2020, 5(6): 5639-5653. doi: 10.3934/math.2020360 |
[8] | Rizwan Gul, Muhammad Shabir, Tareq M. Al-shami, M. Hosny . A Comprehensive study on $ (\alpha, \beta) $-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation. AIMS Mathematics, 2023, 8(11): 25888-25921. doi: 10.3934/math.20231320 |
[9] | Rukchart Prasertpong . Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations. AIMS Mathematics, 2022, 7(2): 2891-2928. doi: 10.3934/math.2022160 |
[10] | Amal T. Abushaaban, O. A. Embaby, Abdelfattah A. El-Atik . Modern classes of fuzzy $ \alpha $-covering via rough sets over two distinct finite sets. AIMS Mathematics, 2025, 10(2): 2131-2162. doi: 10.3934/math.2025100 |
There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely I-δβJ-open sets and I-⋀βJ-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.
Rough set [26,27] is one of a nonstatistical technique to deal with the problems of uncertainty in data and incompleteness of knowledge. The rationale of this set is depended on that the human knowledge is categorized into three fundamental regions, inside, outside and boundary. Therefore, the essential idea of this set focuses on the lower and upper approximations which are used to define the boundary region and accuracy measure. In the classical rough set model approximations are based on the equivalence relations, but this condition does not always hold in many practical problems and also this restriction limits the wide applications of this set. In the recent times, lots of researchers are interested to generalize this set in many fields of applications [9,15,16,23]. It was also generalized by the topological point of view [20,21,29,31] by replacing the equivalence relations in the lower and upper approximations by the open and closed sets, respectively. In the past few years mathematicians turned their attention towards to near (or nearly) open concept as generalization of open sets to topological spaces [1,19,24,25,30]. In this direction, numerous generalizations of the rough set were offered using the nearly open concepts instead of open sets [4,5,6,32]. In 2017, Amer et al. [8] utilized the J-nearly open concepts and introduced the notions of J-nearly approximations. After that, Hosny [11] improved Amer et al.'s approximations [8] by proposing the notions of the δβJ-open sets and ⋀βJ-sets which were used to define the δβJ-approximations and ⋀βJ-approximations.
An ideal is a nonempty collection of sets which is closed under hereditary property and the finite additivity [18,33]. In view of the recent applications of ideals in the rough set theory, it seems very natural to extend the interesting concept of rough set further by using ideals. As, the notions of ideals are pivotal tool helping in removing imprecision and ambiguous of a concept by minimizing the vagueness of uncertainty regions at their borders by increasing the lower approximations and decreasing the upper approximations which automatically implies to increase the accuracy measure of the uncertainty regions [7,13,14]. Recently, Hosny [12] presented the concepts of J-nearly open sets and J-nearly approximations with respect to ideals. She proved that these new sort of J-nearly open sets were generalized the preceding ones [8,11]. Moreover, Hosny's approximations [12] were improvement of Abd El-Monsef et al.'s approximations [2] and Amer et al.'s approximations [8]. Furthermore, the J-nearly rough membership relations and functions with respect to ideals were introduced in [12] as generalization of the other types [3,22,28].
This work indicates that the rough set has a purely topological nature and emphasizes the importance of using ideal in the study of this set because it demystifies the concept. So, a more general notion of a topological rough set via ideal is suggested. In this paper, Section 2 covers some fundamental principles of concepts which are needed in the current work. Meantime, Sections 3 and 5 introduce and study new J-near open sets with respect to ideals namely, I-δβJ-open sets and I-⋀βJ-sets. The basic properties, characterizations and the relationships among of these definitions are derived. These definitions are more general than the previous ones [8,11,12]. It should be noted that the generalization of I-βJ-open sets [12] by using the I-⋀β-sets is very different from the generalization of the I-βJ-open sets by using the I-δβJ-open sets. The main difference is that the family of all I-δβJ-open sets does not form a topology, as the intersection of two I-δβJ-open sets does not need to be an I-δβJ-open set as shown in Example 3.1. While, the family of all I-⋀β-sets forms a topology as it is shown in Lemma 5.2. Moreover, it is shown that the concepts of I-δβJ-open sets and I-⋀β-sets are independent (see Remark 5.5). Furthermore, if I={ϕ}, then the current definitions are coincided with Hosny's definitions [11]. So, Hosny's definitions [11] are special case of the current definitions. The main object of Sections 4 and 6 is to propose two different and independent of new approximations. These approximations are based on I-δβJ-open sets and I-⋀β-sets. The properties of the present approximations and the connections among them are established and constructed in these sections. They are compared to the prior ones [2,8,11,12] and shown that the accuracy measure which deduced by the current approximations is the best. The goal of Section 7 is to define new kind of the rough membership functions via ideal namely, I-δβJ-rough membership functions and I-⋀βJ-rough membership functions. It is proved that these functions are better than the previous ones such as Abd El-Monsef et al. [3], Hosny [12], Lin [22], Pawlak and Skowron [28] (see Lemmas 7.2, 7.3 and Remark 7.8). Section 8 demonstrates the importance of this paper by some real life applications. Finally, Section 9 aims to outline the essential findings and a plan for the future work.
Definition 2.1. [17] Let X be a non-empty set. I≠ϕ, I∈P(X) is an ideal on X, if
(i) A∈I and B∈I⇒A∪B∈I.
(ii) A∈I and B⊆A⇒B∈I.
Definition 2.2. [2] Let X be a non-empty finite set and R be an arbitrary binary relation on X. The J-neighborhood of x∈X (J-nd) (nJ(x)),J∈{R,L,<R>,<L>,I,U,<I>,<U>} defined as:
(i) R-nd: nR(x)={y∈X:xRy}.
(ii) L-nd: nL(x)={y∈X:yRx}.
(iii) <R>-nd: n<R>(x)=∩x∈nR(y)nR(y).
(iv) <L>-nd: n<L>(x)=∩x∈nL(y)nL(y).
(v) I-nd: nI(x)=nR(x)∩nL(x).
(vi) U-nd: nU(x)=nR(x)∪nL(x).
(vii) <I>-nd: n<I>(x)=n<R>(x)∩n<I>(x).
(viii) <U>-nd: n<U>(x)=n<R>(x)∪n<I>(x).
From the following concepts and throughout this paper J∈{R,L,<R>,<L>,I,U,<I>,<U>}.
Definition 2.3. [2] Let X be a non-empty finite set, R be an arbitrary binary relation on X and ΞJ:X→P(X) assigns each x in X its J-nd in P(X). (X,R,ΞJ) is a J-neighborhood space (J-ndS).
Theorem 2.1. [2] Let (X,R,ΞJ) be a J-ndS and A⊆X. Then, τJ={A⊆X:∀a∈A,nJ(a)⊆A} is a topology on X. The elements of τJ are called J-open set and the complement of J-open set is J-closed set. The family ΓJ of all J-closed sets defined by ΓJ={F⊆X:F′∈τJ},F′ is the complement of F.
Definition 2.4. [2] Let (X,R,ΞJ) be a J-ndS and A⊆X. The J-lower, J-upper approximations, J-boundary regions and J-accuracy of A are defined respectively by:
R_J(A) is the union of all J-open sets which are subset of A=intJ(A), where intJ(A) represents J-interior of A.
¯RJ(A) is the intersection of all J-closed sets which are superset of A=clJ(A), where clJ(A) represents J-closure of A.
BNDJ(A)=¯RJ(A)−R_J(A).
ACCJ(A)=|R_J(A)||¯RJ(A)|, where |¯RJ(A)|≠0.
Definition 2.5. [2] Let (X,R,ΞJ) be a J-ndS. A⊆X is J-exact if ¯RJ(A)=R_J(A). Otherwise, A is J-rough.
Definition 2.6. [8] Let (X,R,ΞJ) be a J-ndS. A⊆X is
(i) J-preopen (PJ-open), if intJ(clJ(A))⊇A.
(ii) J-semiopen (SJ-open), if clJ(intJ(A))⊇A.
(iii) αJ-open, if A⊆intJ[clJ(intJ(A))].
(iv) βJ-open (semi preopen), if A⊆clJ[intJ(clJ(A))].
These sets are called J-nearly open sets, the families of J-nearly open sets of X denoted by ηJO(X), the complements of the J-nearly open setsare called J-nearly closed sets and the families of J-nearly closed sets of X denoted by ηJC(X), ∀η∈{P,S,α,β}.
Remark 2.1. [8] The implications between τJ,ΓJ,ηJO(X) and ηJC(X) are in Figure 1.
From the following concepts and throughout this paper η∈{P,S,α,β}.
Definition 2.7. [8] Let (X,R,ΞJ) be a J-ndS and A⊆X. The J-nearly lower, J-nearly upper approximations, J-nearly boundary regions and J-nearly accuracy of A are defined respectively by:
R_ηJ(A) is the union of all J-nearly open sets which are subset of A=J-nearly interior of A.
¯RηJ(A) is the intersection of all J-nearly closed sets which are superset of A=J-nearly closure of A.
BNDηJ(A)=¯RηJ(A)−R_ηJ(A).
ACCηJ(A)=|R_ηJ(A)||¯RηJ(A)|, where |¯RηJ(A)|≠0,|¯RηJ(A)| denotes to the cardinality of ¯RηJ(A).
Definition 2.8. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. The δ-J-closure of A is defined by clδJ(A)={x∈X:A∩intJ(clJ(G))≠ϕ,G∈τJ and x∈G}. A set A is called δJ-closed if A=clδJ(A). The complement of a δJ-closed set is δJ-open. Notice that intδJ(A)=X−clδJ(X−A).
Definition 2.9. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. A subset A is called δβJ-open, if A⊆clJ[intJ(clδJ(A))]. The complement of a δβJ-open set is a δβJ-closed set. The family of all δβJ-open and δβJ-closed are denoted by δβJO(X) and δβJC(X) respectively.
Proposition 2.1. [11] Every βJ-open is δβJ-open.
Definition 2.10. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X Then, the δβJ-lower, δβJ-upper approximations, δβJ-boundary and δβJ-accuracy of A are defined respectively by:
R_δβJ(A)=∪{G∈δβJO(X):G⊆A}=δβJ-interior of A.
¯RδβJ(A)=∩{H∈δβJC(X):A⊆H}=δβJ-closure of A.
BNDδβJ(A)=¯RδβJ(A)−R_δβJ(A).
ACCδβJ(A)=|R_δβJ(A)||¯RδβJ(A)|, where |¯RδβJ(A)|≠0.
Theorem 2.2. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. Then
(i) R_αJ(A)⊆R_pJ(A)⊆R_γJ(A)⊆R_βJ(A)⊆R_δβJ(A).
(ii) R_αJ(A)⊆R_sJ(A)⊆R_γJ(A)⊆R_βJ(A)⊆R_δβJ(A).
(iii) R_J(A)⊆R_δβJ(A).
(iv) ¯RδβJ(A)⊆¯RβJ(A)⊆¯RγJ(A)⊆¯RpJ(A)⊆¯RαJ(A).
(v) ¯RδβJ(A)⊆¯RβJ(A)⊆¯RγJ(A)⊆¯RsJ(A)⊆¯RαJ(A).
(vi) ¯RδβJ(A)⊆¯RJ(A).
Corollary 2.1. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. Then
(i) BNDδβJ(A)⊆BNDβJ(A)⊆BNDγJ(A)⊆BNDpJ(A)⊆BNDαJ(A).
(ii) BNDδβJ(A)⊆BNDβJ(A)⊆BNDγJ(A)⊆BNDsJ(A)⊆BNDαJ(A).
(iii) BNDδβJ(A)⊆BNDJ(A).
(iv) ACCαJ(A)⩽ACCpJ(A)⩽ACCγJ(A)⩽ACCβJ(A)⩽ACCδβJ(A).
(v) ACCαJ(A)⩽ACCsJ(A)⩽ACCγJ(A)⩽ACCβJ(A)⩽ACCδβJ(A).
(vi) ACCJ(A)⩽ACCδβJ(A).
Definition 2.11. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. A subset A is called
(i) δβJ-definable (δβJ-exact) if ¯RδβJ(A)=R_δβJ(A) or BNDδβJ(A)=ϕ.
(ii) δβJ-rough if ¯RδβJ(A)≠R_δβJ(A) or BNDδβJ(A)≠ϕ.
Definition 2.12. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. A subset ⋀βJ is defined as follows: ⋀βJ(A)=∩{G:A⊆G,G∈βJO(X)}. The complement of ⋀βJ(A)-set is called ⋁βJ(A)-set.
Definition 2.13. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. A subset A is called ⋀βJ-set if A=⋀βJ(A). The family of all ⋀βJ-set and ⋁βJ-set are denoted by τ⋀βJ and Γ⋁βJ respectively.
Proposition 2.2. [11] Every βJ-open set is ⋀βJ-set.
Definition 2.14. [11] Let (X,R,ΞJ) be a J-ndS, A⊆X. The ⋀βJ-lower, ⋀βJ-upper approximations, ⋀βJ-boundary and ⋀βJ-accuracy of A are defined respectively by:
R_⋀βJ(A)=∪{G∈τ⋀βJ:G⊆A}=⋀βJ-interior of A.
¯R⋀βJ(A)=∩{H∈Γ⋁βJ:A⊆H}=⋀βJ-closure of A.
BND⋀βJ(A)=¯R⋀βJ(A)−R_⋀βJ(A).
ACC⋀βJ(A)=|R_⋀βJ(A)||¯R⋀βJ(A)|, where |¯R⋀βJ(A)|≠0.
Theorem 2.3. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. Then
(i) R_αJ(A)⊆R_pJ(A)⊆R_γJ(A)⊆R_βJ(A)⊆R_⋀βJ(A).
(ii) R_αJ(A)⊆R_sJ(A)⊆R_γJ(A)⊆R_βJ(A)⊆R_⋀βJ(A).
(iii) R_J(A)⊆R_⋀βJ(A).
(iv) ¯R⋀βJ(A)⊆¯RβJ(A)⊆¯RγJ(A)⊆¯RpJ(A)⊆¯RαJ(A).
(v) ¯R⋀βJ(A)⊆¯RβJ(A)⊆¯RγJ(A)⊆¯RsJ(A)⊆¯RαJ(A).
(vi) ¯R⋀βJ(A)⊆¯RJ(A).
Corollary 2.2. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. Then
(i) BND⋀βJ(A)⊆BNDβJ(A)⊆BNDγJ(A)⊆BNDpJ(A)⊆BNDαJ(A).
(ii) BND⋀βJ(A)⊆BNDβJ(A)⊆BNDγJ(A)⊆BNDsJ(A)⊆BNDαJ(A).
(iii) BND⋀βJ(A)⊆BNDJ(A).
(iv) ACCαJ(A)⩽ACCpJ(A)⩽ACCγJ(A)⩽ACCβJ(A)⩽ACC⋀βJ(A).
(v) ACCαJ(A)⩽ACCsJ(A)⩽ACCγJ(A)⩽ACCβJ(A)⩽ACC⋀βJ(A).
(vi) ACCJ(A)⩽ACC⋀βJ(A).
Definition 2.15. [11] Let (X,R,ΞJ) be a J-ndS and A⊆X. A subset A is called
(i) ⋀βJ-definable (⋀βJ-exact) if ¯R⋀βJ(A)=R_⋀βJ(A) or BND⋀βJ(A)=ϕ.
(ii) ⋀βJ-rough if ¯R⋀βJ(A)≠R_⋀βJ(A) or BND⋀βJ(A)≠ϕ.
Definition 2.16. [12] Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. A⊆X is called
(i) I-αJ-open, if ∃G∈τJ such that (A−intJ(clJ((G))∈I and (G−A)∈I.
(ii) I-J-Preopen (briefly I-PJ-open), if ∃G∈τJ such that (A−G)∈I and (G−clJ(A))∈I.
(iii) I-J-Semi open (briefly I-SJ-open), if ∃G∈τJ such that (A−clJ(G))∈I and (G−A)∈I.
(iv) I-βJ-open, if ∃G∈τJ such that (A−clJ(G))∈I and (G−clJ(A))∈I.
These sets are called I-J-nearly open sets, the complement of the I-J-nearly open sets is called I-J-nearly closed sets, the families of I-J-nearly open sets of X denoted by I-ηJO(X) and the families of I-J-nearly closed sets of X denoted by I-ηJC(X), ∀η∈{P,S,α,β}.
Proposition 2.3. [12] Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:
I−αJ−openI−PJ−open⇓⇓I−SJ−open⇒I−βJ−open. |
Proposition 2.4. [12] Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:
τJ(ΓJ)⇒I−αJO(I−αJC)I−PJO(I−PJC)⇓⇓I−SJO(I−SJC)⇒I−βJO(I−βJC). |
Definition 2.17. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. The I-J-nearly lower, I-J-nearly upper approximations, I-J-nearly boundary regions and I-J-nearly accuracy of A are defined respectively by:
R_I−ηJ(A)=∪{G∈I-ηJO(X):G⊆A}=I-J-nearly interior of A.
¯RI−ηJ(A)=∩{H∈I-ηJC(X):A⊆H}=I-J-nearly closure of A.
BNDI−ηJ(A)=¯RI−ηJ(A)−R_I−ηJ(A).
ACCI−ηJ(A)=|R_I−ηJ(A)||¯RI−ηJ(A)|, where |¯RI−ηJ(A)|≠0.
Definition 2.18. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. A is an I-ηJ-nearly definable (I-ηJ-nearly exact) set if ¯RI−ηJ(A)=R_I−ηJ(A). Otherwise, A is an I-ηJ-nearly rough set.
Theorem 2.4. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) R_ηJ(A)⊆R_I−ηJ(A).
(ii) R_J(A)⊆R_I−ηJ(A).
(iii) ¯RI−ηJ(A)⊆¯RηJ(A).
(iv) ¯RI−ηJ(A)⊆¯RJ(A).
Corollary 2.3. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) BNDI−ηJ(A)⊆BNDηJ(A).
(ii) BNDηJ(A)⊆BNDJ(A).
(iii) ACCηJ(A)⩽ACCI−ηJ(A).
(iv) ACCJ(A)⩽ACCI−ηJ(A).
Proposition 2.5. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) R_I−PJ(A)⊆R_I−βJ(A).
(ii) R_I−αJ(A)⊆R_I−SJ(A)⊆R_I−βJ(A).
(iii) ¯RI−βJ(A)⊆¯RI−PJ(A).
(iv) ¯RI−βJ(A)⊆¯RI−SJ(A)⊆¯RI−αJ(A).
Corollary 2.4. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) BNDI−βJ(A)⊆BNDI−PJ(A).
(ii) BNDI−βJ(A)⊆BNDI−SJ(A)⊆BNDI−αJ(A).
(iii) ACCI−PJ(A)⩽ACCI−βJ(A).
(iv) ACCI−αJ(A)⩽ACCI−SJ(A)⩽ACCI−βJ(A).
Definition 2.19. [28] Let R be an equivalence relation on X and A⊆X. Then the rough membership functions of A⊆X are defined as μA:X→[0,1], where
μA(x)=|[x]R∩A||[x]R|,x∈X. |
[x]R denotes to an equivalence classes.
Definition 2.20. [3] Let (X,R,ΞJ) be a J-ndS, A⊆X and x∈X. Then the J-rough membership functions of A are defined by μJA→[0,1], where
μJA(x)=|{∩nJ(x)}∩A||∩nJ(x)|. |
Definition 2.21. [3] Let (X,R,ΞJ) be a J-ndS, A⊆X and x∈X. Then the J-rough nearly membership functions of A are defined by μηJA→[0,1], where
μηJA(x)={1if1∈ψηJA(x).min(ψηJA(x))otherwise.}.andψηJA(x)=|ηJ(x)∩A||ηJ(x)|,x∈ηJ(x),ηJ(x)∈ηJO(X). |
Definition 2.22. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X,A⊆X and x∈X. The I−J-nearly rough membership functions of A are defined by μI−ηJA→[0,1], where
μI−ηJA(x)={1if1∈ψI−ηJA(x).min(ψI−ηJA(x))otherwise.}.andψI−ηJA(x)=|I−ηJ(x)∩A||I−ηJ(x)|,x∈I−ηJ(x),I−ηJ(x)∈I−ηJO(X). |
Lemma 2.1. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) μJA(x)=1⇒μηJA(x)=1⇒μI−ηJA(x)=1,∀x∈X.
(ii) μJA(x)=0⇒μηJA(x)=0⇒μI−ηJA(x)=0,∀x∈X.
Definition 2.23. [3] Let (X,R,ΞJ) be a J-ndS, x∈X and A⊆X:
(i) If x∈R_J(A), then x is J-surely belongs to A, denoted by x∈_JA.
(ii) If x∈¯RJ(A), then x is J-possibly belongs to A, denoted by x¯∈JA.
(iii) If x∈R_ηJ(A), then x is J-nearly surely (ηJ-surely) belongs to A, denoted by x∈_ηJA.
(iv) If x∈¯RηJ(A), then x is J-nearly possibly (ηJ-possibly) belongs to A, denoted by x¯∈ηJA.
It is called J-(nearly) strong and J-(nearly) weak membership relations respectively.
Definition 2.24. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X,x∈X and A⊆X:
(i) If x∈R_I−ηJ(A), then x is J-nearly surely with respect to I (I−ηJ-surely) belongs to A, denoted by x∈_I−ηJA.
(ii) If x∈¯RI−ηJ(A), then x is J-nearly possibly with respect to I (briefly I−ηJ-possibly) belongs to A, denoted by x¯∈I−ηJA.
It is called J-nearly strong and J-nearly weak membership relations with respect to I respectively.
Proposition 2.6. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then
(i) If x∈_JA⇒x∈_ηJA⇒x∈_I−ηJA.
(ii) If x¯∈I−ηJA⇒x¯∈ηJA⇒x¯∈JA.
In this section, the concept of I-δβJ-open sets is presented as generalization of the J-nearly open sets in Definitions 2.6 [8], 2.9 [11] and also generalization of the I-J-nearly open sets in Definition 2.16 [12]. This concept is based on the notions of ideals. Moreover, the principle properties of this concept is studied and compared to the previous concepts.
Definition 3.1. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. A⊆X is called I-δβJ-open, if ∃G∈τJ such that (A−clJ(G))∈I and (G−clδJ(A))∈I. The complement of the I-δβJ-open sets is called I-δβJ-closed sets. The family of all I-δβJ-open and I-δβJ-closed are denoted by I-δβJO(X) and I-δβJC(X) respectively.
Example 3.1. Let
X={a,b,c,d},I={ϕ,{c}}, |
and
R={(a,a),(a,b),(b,a),(b,b),(c,c),(d,a),(d,b),(d,c),(d,d)} |
be a binary relation defined on X, thus aR=bR={a,b},cR={c} and dR=X. Then, the topology associated with this relation is τR={X,ϕ,{c},{a,b},{a,b,c}} and I-δβRO(X)=P(X).
The following proposition shows that the concept of I-δβJ-open sets is an extension of the concept of δβJ-open sets in Definition 2.9 [11].
Proposition 3.1. Every δβJ-open is I-δβJ-open.
Proof. By using Definitions 2.9 [11] and 3.1.
Remark 3.1. (i) The converse of Proposition 3.1 is not necessarily true as shown in Example 3.1, I-δβRO(X)=P(X) and δβRO(X)=P(X)−{{d}}. It is clear that {d} is an I-δβR-open set, but it is not a δβR-open set.
(ii) According to Remark 2.1 [8] and Propositions 2.1 [11], 3.1, the current Definition 3.1 is also a generalization of Definition 2.6 [8].
The following theorem shows that Hosny's Definition 2.9 [11] is a special case of the current definition.
Theorem 3.1. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. If I={ϕ} in the current Definition 3.1, then I get Hosny's Definition 2.9 [11].
Proof. Straightforward.
The following proposition shows that the I-δβJ-open sets are generalization of the I-βJ-open sets [12]. Consequently, they are also generalization of any I-J-near open sets in Definition 2.16 [12] such as, I-PJ-open, I-SJ-open and I-αJ-open sets.
Proposition 3.2. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:
I−αJ−open I−PJ−open ⇓ ⇓I−SJ−open ⇒ I−βJ−open ⇒ I−δβJ−open. |
Proof. Straightforward by Proposition 2.3 [12], Definitions 2.16 [12] and 5.2.
It should be noted that, Proposition 3.2 shows that, every I-βJ-open is I-δβJ-open, but the converse is not necessarily true as shown in the following example.
Example 3.2. Let X={a,b,c},I={ϕ,{b}} and R={(a,a),(a,c),(b,a),(b,c),(c,c)} is a binary relation defined on X thus aR=bR={a,b} and cR={c}. Then, the topology associated with this relation is τR={X,ϕ,{c},{a,b}}. It is clear that {b} is an I-δβR-open set, but it is not an I-βR-open set.
Proposition 3.3. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:
τJ(ΓJ)⇒I−αJO(I−αJC) I−PJO(I−PJC) ⇓ ⇓ I−SJO(I−SJC)⇒I−βJO(I−βJC)⇒I−δβJO(I−δβJC). |
Proof. By Propositions 2.4 and 3.2 [12], the proof is obvious.
Theorem 3.2. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the union of two I-δβJ-open sets is also I-δβJ-open set.
Proof. Let A and B be I-δβJ-open sets. Then, ∃G,H such that (A−clJ(G))∈I,(G−clδJ(A))∈I,(B−clJ(H))∈I and (H−clδJ(B))∈I. Hence, (G−clδJ(A∪B))⊆(G−clδJ(A))∈I, (H−clδJ(A∪B))⊆(H−clδJ(B))∈I and so, (G−clδJ(A∪B))∪(H−clδJ(A∪B))∈I. Let W=G∪H, then (W−clδJ(A∪B))∈I. Also, (A−clJ(W))⊆(A−clJ(G))∈I and (B−clJ(W))⊆(B−clJ(H))∈I. Then, (A−clJ(W))∪(B−clJ(W))⊆(A−clJ(G))∪(B−clJ(H))∈I and so ((A∪B)−clJ(W))⊆(A−clJ(G))∪(B−clJ(H))∈I. Thus, A∪B is an I-δβJ-open set. The rest of the proof is similar.
Remark 3.2. The family of all I-δβJ-open sets in a space X does not form a topology as it is shown in the following example.
Example 3.3. Let
X={a,b,c,d,e},I={ϕ,{c}} |
and
R={(a,a),(a,b),(b,b),(c,c),(c,d),(d,c),(d,d),(e,b),(e,c),(e,d)}. |
It is clear that the intersection of two I-δβR-open sets is not an I-δβR-open set. Take A={a,e} and B={b,e}∈I-δβRO(X), then A∩B={e}∉I-δβRO(X)=P(X)−{{e}}.
Remark 3.3. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A⊆X. Then the following statements are not true in general:
(i) I-δβUO(X)⊆I-δβRO(X)⊆I-δβIO(X).
(ii) I-δβUO(X)⊆I-δβLO(X)⊆I-δβIO(X).
(iii) I-δβ<U>O(X)⊆I-δβ<R>O(X)⊆I-δβ<I>O(X).
(iv) I-δβ<U>O(X)⊆I-δβ<L>O(X)⊆I-δβ<I>O(X).
(v) I-δβRO(X) is the dual of I-δβLO(X).
(vi) I-δβ<R>O(X) is the dual of I-δβ<L>O(X).
So, the relationships among I-δβJ- open sets are not comparable as in Example 3.3:
(i) I-δβRO(X)=P(X)−{{e}}.
(ii) I-δβLO(X)=I-δβ<L>O(X)=P(X)−{{b}}.
(iii) I-δβIO(X)=I-δβUO(X)=I-δβ<R>O(X)=I-δβ<I>O(X)=I-δβ<U>O(X)=P(X).
It is clear that
● I-δβUO(X)⊈- {\delta\beta}_{R}O(X).
● \mathcal{I} - {\delta\beta}_{I}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{R}O(X).
● \mathcal{I} - {\delta\beta}_{U}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{L}O(X).
● \mathcal{I} - {\delta\beta}_{I}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{L}O(X).
● \mathcal{I} - {\delta\beta}_{ < U > }O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{ < L > }O(X).
● \mathcal{I} - {\delta\beta}_{ < I > }O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{ < L > }O(X).
● \mathcal{I} - {\delta\beta}_{R}O(X) is not the dual of \mathcal{I} - {\delta\beta}_{L}O(X) and \mathcal{I} - {\delta\beta}_{ < R > }O(X) is not the dual of \mathcal{I} - {\delta\beta}_{ < L > }O(X).
● In a similar way, I can add examples to show that \mathcal{I} - {\delta\beta}_{L}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{I}O(X), \mathcal{I} - {\delta\beta}_{R}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{U}O(X), \mathcal{I} - {\delta\beta}_{R}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{I}O(X), \mathcal{I} - {\delta\beta}_{L}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{U}O(X), \mathcal{I} - {\delta\beta}_{I}O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{L}O(X), \mathcal{I} - {\delta\beta}_{ < L > }O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{ < I > }O(X), \mathcal{I} - {\delta\beta}_{ < L > }O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{ < U > }O(X), \mathcal{I} - {\delta\beta}_{ < R > }O(X) \nsubseteq \mathcal{I} - {\delta\beta}_{ < I > }O(X) and \mathcal{I} - {\delta\beta}_{ < U > }O(X)\nsubseteq \mathcal{I} - {\delta\beta}_{ < R > }O(X).
The purpose of this section is to generalize the previous approximations in Definitions 2.4 [2], 2.7 [8], 2.10 [11] and 2.17 [12]. The current approximations are depended on the \mathcal{I} - {\delta\beta}_{J} -open sets. The fundamental properties of these approximations are obtained. Furthermore, the current findings are compared to the previous approaches.
Definition 4.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. The \mathcal{I} - {\delta\beta}_{J} -lower, \mathcal{I} - {\delta\beta}_{J} -upper approximations, \mathcal{I} - {\delta\beta}_{J} -boundary regions and \mathcal{I} - {\delta\beta}_{J} -accuracy of A are defined respectively by:
{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = \cup\{G \in {\mathcal{I}} - {\delta\beta}_JO(X) : G \subseteq A\} = \mathcal{I} - {\delta\beta}_{J} -interior of A .
{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = \cap\{H \in {\mathcal{I}} - {\delta\beta}_JC(X): A \subseteq H\} = \mathcal{I} - {\delta\beta}_{J} -closure of A .
{{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)-{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
{{ACC}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = \frac{|{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)|}{|{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)|}, where |{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)|\neq0.
The following proposition presents the main properties of the current \mathcal{I} - {\delta\beta}_{J} -lower and \mathcal{I} - {\delta\beta}_{J} -upper approximations.
Proposition 4.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A, B \subseteq X. Then,
(i) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq A\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) equality hold if A = \phi or X.
(ii) A\subseteq B\Rightarrow {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(iii) A\subseteq B\Rightarrow {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A\cap B)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\cap{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(v) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A\cup B)\supseteq{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\cup{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(vi) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A\cup B)\supseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\cup{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(vii) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A\cap B)\subseteq{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\cap{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(B).
(viii) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = ({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A^{'}))^{'} , {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = ({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A^{'}))^{'} .
(ix) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)) = {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(x) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)) = {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(xi) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A))\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)).
(xii) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A))\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)).
(xiii) x\in{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\Leftrightarrow G\cap A\neq\phi, \forall G\in \mathcal{I} - {{\delta\beta}}_{J}O(X), x\in G.
(xiv) x\in{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\Leftrightarrow \exists \; G\in \mathcal{I} - {\delta\beta}_{J}O(X), x\in G, G\subseteq A.
The proof of this proposition is simple using the \mathcal{I} - {\delta\beta}_{J} -interior and \mathcal{I} - {\delta\beta}_{J} -closure, so I omit it.
Remark 4.1. Example 3.3 shows that
(a) the inclusion in Proposition 4.1 parts (i), (iv), (v), (vi), (vii), (xi) and (xii) can not be replaced by equality relation:
(i) For part (i), if A = \{a, b, c, d\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = X, then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\nsubseteq A, take A = \{e\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = \phi . Then, A\nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A).
(ii) For part (iv), if A = \{a, b, c, d\}, B = \{b, c, d, e\}, A\cap B = \{b, c, d\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = X, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = B, {\overline{\mathcal{R}}}^{\delta\beta}_{R}(A\cap B) = A\cap B , then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\cap {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = \{b, c, d, e\}\nsubseteq \{b, c, d\} = {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cap B).
(iii) For part (v), if A = \{a\}, B = \{e\}, A\cup B = \{a, e\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = A, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = \phi, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cup B) = A\cup B , then {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cup B) = \{a, e\} \nsubseteq \{a\} = {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\cup {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B).
(iv) For part (vi), if A = \{a, c\}, B = \{b, d\}, A\cup B = \{a, b, c, d\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = A, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = B, \\ {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cup B) = X , then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cup B) = X\nsubseteq \{a, b, c, d\} = {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\cup {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B).
(v)For part (vii), if A = \{a, e\}, B = \{c, e\}, A\cap B = \{e\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = A, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = B, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cap B) = \phi , then {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\cap {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = \{e\}\nsubseteq \phi = {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A\cap B).
(vi) For part (xi), if A = \{a, b, c, d\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) = A, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) = X , then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) \nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)).
(vii) For part (xii), if A = \{e\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) = A, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) = \phi , then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)) \nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}({\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)).
(b) the converse of parts (ii) and (iii) is not necessarily true:
(i) For part (ii), if A = \{e\}, B = \{a, b, c, d\}, then {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = A, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = X . Therefore, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) , but A\nsubseteq B.
(ii) For part (iii), if A = \{e\}, B = \{c, d\}, then {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = \phi, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) = B . Therefore, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(B) , but A\nsubseteq B.
Definition 4.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. A is an \mathcal{I} - {\delta\beta}_{J} -definable (an \mathcal{I} - {\delta\beta}_{J} -exact) set if {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A). Otherwise, A is an \mathcal{I} - {\delta\beta}_{J} -rough set.
In Example 3.3 A = \{c\} is \mathcal{I} - {\delta\beta}_{R} -exact, while B = \{e\} is \mathcal{I} - {\delta\beta}_{R} -rough.
Remark 4.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then the intersection of two \mathcal{I} - {\delta\beta}_{R} -exact sets does not need to be an \mathcal{I} - {\delta\beta}_{R} -exact set as in Example 3.3 \{a, e\} and \{c, e\} are \mathcal{I} - {\delta\beta}_{R} -exact sets, but \{a, e\}\cap\{c, e\} = \{e\} is not an \mathcal{I} - {\delta\beta}_{R} -exact set.
The following theorem and corollary present the relationships between the current approximations in Definition 4.1 and the previous ones in Definitions 2.4 [2], 2.7 [8] and 2.10 [11].
Theorem 4.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS and A \subseteq X. Then
(i) {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{p}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(ii) {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{s}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(iii) {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq{\overline{\mathcal{R}}}^{\delta\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{p}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\alpha}_{J}(A).
(v) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq{\overline{\mathcal{R}}}^{\delta\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{s}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\alpha}_{J}(A).
(vi) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A)\subseteq {\overline{\mathcal{R}}}_{J}(A).
Proof. (i) By Theorem 2.2 [11], {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{p}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A), and {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A) = \cup\{G \in {\delta\beta}_JO(X):G \subseteq A\}\subseteq \cup\{G \in {\mathcal{I}} - {\delta\beta}_JO(X) : G \subseteq A\} = {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) (by Proposition 3.1).
(ii) It is similar to (i).
(iii) By Theorem 2.2 [11], {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A), and by (1) {\underline{\mathcal{R}}}^{\delta\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A). Hence, {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(iv)–(vi) They are similar to (i)–(iii).
Corollary 4.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS and A \subseteq X. Then
(i) {\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}(A)\subseteq{\mathcal{BND}}^{\delta\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\gamma}_{J}(A)\subseteq {\mathcal{BND}}^{p}_{J}(A)\subseteq {\mathcal{BND}}^{\alpha}_{J}(A).
(ii) {\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}(A)\subseteq{\mathcal{BND}}^{\delta\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\gamma}_{J}(A)\subseteq {\mathcal{BND}}^{s}_{J}(A)\subseteq {\mathcal{BND}}^{\alpha}_{J}(A) .
(iii) {\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}(A)\subseteq {\mathcal{BND}}_{J}(A).
(iv) {ACC}^{\alpha}_{J}(A)\leqslant {ACC}^{p}_{J}(A)\leqslant {ACC}^{\gamma}_{J}(A)\leqslant {ACC}^{\beta}_{J}(A)\leqslant {ACC}^{\delta\beta}_{J}(A)\leqslant {ACC}^{\mathcal{I}-{\delta\beta}}_{J}(A).
(v) {ACC}^{\alpha}_{J}(A)\leqslant {ACC}^{s}_{J}(A)\leqslant {ACC}^{\gamma}_{J}(A)\leqslant {ACC}^{\beta}_{J}(A)\leqslant {ACC}^{\delta\beta}_{J}(A)\leqslant {ACC}^{\mathcal{I}-{\delta\beta}}_{J}(A).
(vi) {ACC}_{J}(A)\leqslant {ACC}^{\mathcal{I}-{\delta\beta}}_{J}(A).
Remark 4.3. Example 3.1 shows that the converse of the implications in Theorem 4.1 and Corollary 4.1 is not true in general. Take A = \{d\}, then {\underline{\mathcal{R}}}_{R}(A) = {\underline{\mathcal{R}}}^{\delta\beta}_{R}(A) = \phi, {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = \{d\} and if A = \{a, b, c\}, then {\overline{\mathcal{R}}}_{R}(A) = {\overline{\mathcal{R}}}^{\delta\beta}_{R}(A) = X, {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = \{a, b, c\}. Moreover, take A = \{a, b, c\}, then the boundary and accuracy by the present method in Definition 4.1 are \phi and 1 respectively. Whereas, the boundary and accuracy by using Abd El-Monsef et al.'s method 2.4 [2], Amer et al.'s method 2.7 [8] and Hosny's method 2.10 [11] are \{d\} and 0 respectively.
Corollary 4.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) A is \alpha_J -exact \Rightarrow A is S_J -exact \Rightarrow A is \beta_J -exact \Rightarrow {\delta\beta}_J -exact \Rightarrow A is \mathcal{I} - {\delta\beta}_J -exact.
(ii) A is P_J -exact \Rightarrow A is \beta_J -exact \Rightarrow A is {\delta\beta}_J -exact \Rightarrow A is \mathcal{I} - {\delta\beta}_J -exact.
(iii) A is {J} -exact \Rightarrow A is \mathcal{I} - {\delta\beta}_{J} -exact.
(iv) A is \mathcal{I} - {\delta\beta}_{J} -rough \Rightarrow A is {\delta\beta}_J -rough \Rightarrow A is {\beta}_{J} -rough \Rightarrow A is S_J -rough \Rightarrow A is \alpha_J -rough.
(v) A is \mathcal{I} - {\delta\beta}_{J} -rough \Rightarrow A is {\delta\beta}_J -rough \Rightarrow A is {\beta}_{J} -rough \Rightarrow A is P_J -rough.
(vi) A is \mathcal{I} - {\delta\beta}_{J} -rough \Rightarrow A is {J} -rough.
Remark 4.4. The converse of parts of Corollary 4.2 is not necessarily true as in Example 3.1:
(i) If A = \{d\} , then it is \mathcal{I} - {\delta\beta}_{R} -exact, but it is neither {\delta\beta}_{R} -exact nor {R} -exact.
(ii) If A = \{a, b, c\} , then it is {R} -rough and {\delta\beta}_{R} -rough, but it is not \mathcal{I} - {\delta\beta}_{R} -rough.
The following proposition and corollary are introduced the relationships between the current approximations in Definition 4.1 and the previous one in Definition 2.17 [12].
Proposition 4.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {\underline{\mathcal{R}}^{\mathcal{I}-P}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A).
(ii) {\underline{\mathcal{R}}^{\mathcal{I}-\alpha}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-S}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) .
(iii) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-P}_J}(A).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-S}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-\alpha}_J}(A).
Proof. By Proposition 3.2, the proof is obvious.
Corollary 4.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\beta}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-P}_J}(A).
(ii) {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\beta}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-S}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\alpha}_J}(A).
(iii) {{ACC}^{\mathcal{I}-P}_J}(A)\leqslant {{ACC}^{\mathcal{I}-\beta}_J}(A)\leqslant {{ACC}^{\mathcal{I}-{\delta\beta}}_J}(A) .
(iv) {{ACC}^{\mathcal{I}-\alpha}_J}(A)\leqslant {{ACC}^{\mathcal{I}-S}_J}(A)\leqslant {{ACC}^{\mathcal{I}-\beta}_J}(A)\leqslant {{ACC}^{\mathcal{I}-{\delta\beta}}_J}(A).
Corollary 4.4. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) A is \mathcal{I} - \alpha_J -exact \Rightarrow A is \mathcal{I} - S_J -exact \Rightarrow A is \mathcal{I} - \beta_J -exact \Rightarrow A is \mathcal{I} - {\delta\beta}_J -exact.
(ii) A is \mathcal{I} - P_J -exact \Rightarrow A is \mathcal{I} - \beta_J -exact \Rightarrow A is \mathcal{I} - {\delta\beta}_J -exact.
(iii) A is \mathcal{I} - {\delta\beta}_{J} -rough \Rightarrow A is \mathcal{I} - {\beta}_{J} -rough \Rightarrow A is \mathcal{I} - S_J -rough \Rightarrow A is \mathcal{I} - \alpha_J -rough.
(iv) A is \mathcal{I} - {\delta\beta}_{J} -rough \Rightarrow A is \mathcal{I} - {\beta}_{J} -rough \Rightarrow A is \mathcal{I} - P_J -rough.
In Table 1, the lower, upper approximations, boundary regions and accuracy are calculated by using Hosny's approximations 2.17 [12] and the current approximations in Definition 4.1 by using Example 3.2.
{A} | The previous one in Definition 2.17 [12] | The current method in Definition 2.17 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\delta\beta}}_R}(A) | |
\{a\} | \phi | \{a\} | \{a\} | 0 | \{a\} | \{a\} | \phi | 1 |
\{b\} | \phi | \{b\} | \{b\} | 0 | \{b\} | \{b\} | \phi | 1 |
\{c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{c\} | \{c\} | \phi | 1 |
\{a, b\} | \phi | \{a, b\} | \{a, b\} | 0 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{a, c\} | \{a, c\} | \phi | 1 |
\{b, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{b, c\} | \{b, c\} | \phi | 1 |
X | X | X | \phi | 1 | X | X | \phi | 1 |
For example, take A = \{a, b\}, then the boundary and accuracy by the present method in Definition 4.1 are \phi and 1 respectively. Whereas, the boundary and accuracy by using Hosny's method 2.17 [12] are \{a, b\} and 0 respectively.
Remark 4.5. Example 3.2 shows that the converse of the implications in Corollary 4.4 is not true in general. For example, if take A = \{a\}, then it is \mathcal{I} - {\delta\beta}_R -exact, but it is not \mathcal{I} - \beta_R -exact and consequently, not \mathcal{I} - S_R -exact, not \mathcal{I} - \alpha_R -exact and not \mathcal{I} - P_R -exact, also A = \{a\}, is \mathcal{I} - \beta_R -rough, but not \mathcal{I} - {\delta\beta}_R -rough.
Remark 4.6. Theorem 4.1 and Proposition 4.2 show that the present method in Definition 4.1 reduces the boundary region by increasing the \mathcal{I} - {\delta\beta}_{J} -lower approximations and decreasing the \mathcal{I} - {\delta\beta}_{J} -upper approximations with the comparison of Abd El-Monsef et al.'s method 2.4 [2], Amer et al.'s method 2.7 [8], Hosny's method 2.10 [11] and Hosny's method 2.17 [12]. Moreover, Corollaries 4.1 and 4.3 show that the current accuracy in Definition 4.1 is greater than the previous ones in Definitions 2.4 [2], 2.7 [8], 2.10 [11] and 2.17 [12].
The idea of generalization of J -nearly open sets and \mathcal{I} - J -nearly open sets is developed and extended in this section by proposing the concept of \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets. The main characterizations of this concept and the connections among them are investigated and analyzed. The concepts of \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets and \mathcal{I} - {\delta\beta}_{J} -open sets are different and independent (see Remark 5.5).
Definition 5.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X . A subset \mathcal{I} - {\bigwedge_{\beta}}_{J} is defined as follows: \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) = \cap\{ G:A\subseteq G, G\in{\mathcal{I}} - {\beta}_JO(X)\}. The complement of \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) -set is called \mathcal{I} - {\bigvee_{\beta}}_{J}(A) -set.
In the following lemma I summarize the fundamental properties of the subset \mathcal{I} - {\bigwedge_{\beta}}_{J}.
Lemma 5.1. For subsets A, B and A_{\alpha} (\alpha \in \Delta) of a J -ndS (X, \mathcal{R}, \Xi_J), the following implications hold:
(i) A\subseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) .
(ii) If A \subseteq B , then \mathcal{I} - {\bigwedge_{\beta}}_{J}(A)\subseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}(B).
(iii) \mathcal{I} - {\bigwedge_{\beta}}_{J}(\mathcal{I} - {\bigwedge_{\beta}}_{J}(A)) = \mathcal{I} - {\bigwedge_{\beta}}_{J}(A).
(iv) If A \in {\mathcal{I}} - {\beta}_JO(X) , then A = \mathcal{I} - {\bigwedge_{\beta}}_{J}(A). (v) \mathcal{I} - {\bigwedge_{\beta}}_{J}(\cup\{A_{\alpha}:\alpha \in \Delta\}) = \cup\{\mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}):\alpha \in \Delta\}.
(vi) \mathcal{I} - {\bigwedge_{\beta}}_{J}(\cap\{A_{\alpha}:\alpha \in \Delta\})\subseteq\cap\{\mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}):\alpha \in \Delta\}.
Proof. I prove only (v) and (vi) since the other are consequences of Definition 5.1.
(v) First for each \alpha \in \Delta, \mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha})\subseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}({\cup}_{\alpha \in \Delta}A_{\alpha}). Hence, {\cup}_{\alpha \in \Delta} \mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha})\subseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}({\cup}_{\alpha \in \Delta}A_{\alpha}). Conversely, suppose that x\not \in {\cup}_{\alpha \in \Delta} \mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}). Then, x\not \in \mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}) for each \alpha \in \Delta and hence there exists G_{\alpha} \in {\mathcal{I}} - {\beta}_JO(X) such that A_{\alpha} \subseteq G_{\alpha} and x\not\in G_{\alpha} for each \alpha \in \Delta. I have that {\cup}_{\alpha \in \Delta} A_{\alpha} \subseteq {\cup}_{\alpha \in \Delta} G_{\alpha} and {\cup}_{\alpha \in \Delta} G_{\alpha} is {\mathcal{I}} - {\beta}_J -open set which does not contain x. Therefore, x\not\in \mathcal{I} - {\bigwedge_{\beta}}_{J}({\cup}_{\alpha \in \Delta}A_{\alpha}). Thus, \mathcal{I} - {\bigwedge_{\beta}}_{J}({\cup}_{\alpha \in \Delta}A_{\alpha}) \subseteq {\cup}_{\alpha \in \Delta}\mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}).
(vi) Suppose that, x\not\in \cap\{\mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{\alpha}):\alpha \in \Delta\}. There exists \alpha_{0}\in \Delta such that x\not\in \mathcal{I} - {\bigwedge_{\beta}}_{J}(A_{{\alpha}_{0}}) and there exists {\mathcal{I}} - {\beta}_J -open set G such that x\not\in G and A_{{\alpha}_{0}} \subseteq G. I have that {\cap}_{\alpha \in \Delta}A_{{\alpha}}\subseteq A_{{\alpha}_{0}}\subseteq G and x\not\in G. Therefore, x\not\in \mathcal{I} - {\bigwedge_{\beta}}_{J}(\cap\{A_{\alpha}:\alpha \in \Delta\}).
Remark 5.1. The inclusion in Lemma 5.1 parts (i) and (vi) can not be replaced by equality relation. Moreover, the converse of part (ii) is not necessarily true as shown in Example 3.3 that:
(i) For part (i), if A = \{a\}, then \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) = \{a, b\} and \mathcal{I} - {\bigwedge_{\beta}}_{J}(A)\nsubseteq A.
(ii) For part (vi), if A = \{b\} and B = \{a\}, then A\cap B = \phi and \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) = \{b\}, \mathcal{I} - {\bigwedge_{\beta}}_{J}(B) = \{a, b\}, \mathcal{I} - {\bigwedge_{\beta}}_{J}(A\cap B) = \phi and \mathcal{I} - {\bigwedge_{\beta}}_{J}(A)\cap \mathcal{I} - {\bigwedge_{\beta}}_{J}(B) = \{b\}\nsubseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}(A\cap B) = \phi.
(iii) For part (ii), if A = \{a\} and B = \{b\}, then \mathcal{I} - {\bigwedge_{\beta}}_{J}(A) = \{a, b\} and \mathcal{I} - {\bigwedge_{\beta}}_{J}(B) = \{b\} . Therefore, \mathcal{I} - {\bigwedge_{\beta}}(A)\subseteq \mathcal{I} - {\bigwedge_{\beta}}_{J}(B), but A\nsubseteq B.
Definition 5.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X . A subset A is called an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set if A = \mathcal{I}-{\bigwedge_{\beta}}_{J}(A). The family of all \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets and \mathcal{I} - {\bigvee_{\beta}}_{J} -sets are denoted by \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{J} and \Gamma^{\mathcal{I}-{\bigvee_{\beta}}}_{J} respectively.
Example 5.1. In Example 3.1, \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} = P(X), in Example 3.2, \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} = P(X)-\{\{a\}, \{b\}, \{a, b\}\} and in Example 3.2, \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} = \{X, \phi, \{b\}, \{c\}, \{d\}, \{e\}, \{a, b\}, \{b, c\}, \{b, d\}, \{b, e\}, \{ c, d\}, \{c, e\}, \{d, e\}, \{a, b, c\}, \{a, b, \\ e\}, \{a, b, d\}, \{b, c, d\}, \{b, c, e\}, \{b, d, e\}, \{c, d, e\}, \{a, b, c, d\}, \{a, b, c, e\}, \{a, b, d, e\}, \{b, c, d, e\}\}.
The following proposition shows that the concept of \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets is an extension of the concept of {\bigwedge_{\beta}}_{J} -sets.
Proposition 5.1. Every {\bigwedge_{\beta}}_{J} -set is {\mathcal{I}} - {\bigwedge_{\beta}}_{J} -set.
Proof. By using Definitions 2.13 [11] and 5.2.
Remark 5.2. (i) According to Remark 2.1 [8] and Propositions 2.2 [11], 5.1 the current Definition 5.2 is also generalization of any J -near open sets in Definition 2.6 [8] such as, P_J -open, S_J -open and \alpha_J -open sets.
(ii) The converse of Proposition 5.1 is not necessarily true as shown in the following example.
Example 5.2. Let
\begin{align*} X = \{a, b, c, d\}, \mathcal{I} = \{\phi, \{c\}\} \end{align*} |
and
\begin{align*} \mathcal{R} = \{(a, a), (a, c),(b,a),(b,c),(c,c), (d,d)\} \end{align*} |
be a binary relation defined on X thus a\mathcal{R} = b\mathcal{R} = \{a, c\}, c\mathcal{R} = \{c\} and d\mathcal{R} = \{d\}. Then, the topology associated with this relation is \tau_R = \{X, \phi, \{c\}, \{d\}, \{a, c\}, \{c, d\}, \{a, c, d\}\}. It is clear that \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} = P(X) and \tau^{\bigwedge_{\beta}}_{R} = \{X, \phi, \{b\}, \{c\}, \{d\}, \{a, c\}, \{b, c\}, \{b, d\}, \{c, d\}, \{a, b, c\}, \{a, c, d\}, \{b, c, d\}\}.
The following theorem shows that Hosny's Definition 2.13 [11] is a special case of the current definition.
Theorem 5.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS and \mathcal{I} be an ideal on X . If \mathcal{I} = \{\phi\} in the current Definition 5.2, then I get Hosny 's Definition 2.13 [11].
Proof. Straightforward.
The following proposition shows that \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets are generalization of \mathcal{I} - {\beta}_{J} -open sets in Definition 2.16 [12]. Consequently, it is also generalization of any {\mathcal{I}} - J -near open sets in Definition 2.16 [12] such as, {\mathcal{I}} - P_J -open, {\mathcal{I}} - S_J -open and {\mathcal{I}} - \alpha_J -open sets.
Proposition 5.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS and \mathcal{I} be an ideal on X. Then, the following implications hold:
\begin{align} & \mathcal{I}-{{\alpha }_{J}}-open\ \ \ \ \ \ \mathcal{I}-{{P}_{J}}-open \\ & \ \ \ \ \ \Downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Downarrow \\ & \mathcal{I}-{{S}_{J}}-open\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ \mathcal{I}-{{\beta }_{J}}-open\ \ \ \ \ \ \Rightarrow \ \ \ \mathcal{I}-{{\wedge }_{{{\beta }_{J}}}}-set. \\ \end{align} |
Proof. Straightforward by Proposition 2.3 [12], Definitions 2.16 [12] and 29.
Remark 5.3. The converse of Proposition 5.2 is not necessarily true as shown in Example 3.3, \{e\} is an {\mathcal{I}} - {\bigwedge_{\beta}}_{R} -set, but it is not an {\mathcal{I}} - {\beta}_{R} -open set.
Proposition 5.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS and \mathcal{I} be an ideal on X . Then, the following implications hold:
\begin{align} & {{\tau }_{J}}({{\Gamma }_{J}})\Rightarrow \mathcal{I}-{{\alpha }_{J}}O(\mathcal{I}-{{\alpha }_{J}}C)\ \ \ \ \mathcal{I}-{{P}_{J}}O(\mathcal{I}-{{P}_{J}}C) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Downarrow \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{I}-{{S}_{J}}O(\mathcal{I}-{{S}_{J}}C)\Rightarrow \mathcal{I}-{{\beta }_{J}}O(\mathcal{I}-{{\beta }_{J}}C)\Rightarrow \tau _{J}^{\mathcal{I}-{{\wedge }_{\beta }}}(\Gamma _{J}^{\mathcal{I}-{{\vee }_{\beta }}}). \\ \end{align} |
Proof. By Propositions 2.4 [12] and 5.2, the proof is obvious.
In the following lemma I summarize the fundamental properties of {\mathcal{I}} - {\bigwedge_{\beta}}_{J} -sets.
Lemma 5.2. For subsets A, B and A_{\alpha} (\alpha \in \Delta) of a J -ndS (X, \mathcal{R}, \Xi_J), the following implications hold:
(i) X, \phi are \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets.
(ii) If A_{\alpha} is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set \forall \alpha \in \Delta, then {\cup}_{\alpha \in \Delta}A_{\alpha} is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set.
(iii) If A_{\alpha} is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set \forall \alpha \in \Delta, then {\cap}_{\alpha \in \Delta}A_{\alpha} is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set.
Proof. This follows from Lemma 5.1.
Remark 5.4. It is clear from (i)–(iii) in Lemma 5.2 that the family of all \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets forms a topology.
Remark 5.5. The \mathcal{I} - {\delta\beta}_{J} -open sets of Definition 3.1 and the current Definition 5.2 of \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets are different and independent. Example 3.3 shows that \{a\} is an \mathcal{I} - {\delta\beta}_{J} -open set, but it is not an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set. Moreover, it shows that \{e\} is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -set, but it is not an \mathcal{I} - {\delta\beta}_{J} -open set.
Remark 5.6. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then the following statements are not true in general:
(i) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U} \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}.
(ii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U} \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L} \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}.
(iii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > } \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > } \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }.
(iv) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > } \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > } \subseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }.
(v) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} is the dual of \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L}.
(vi) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > } is the dual of \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }.
So, the relationships among \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets are not comparable as in Example 3.3:
(i) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} = \{X, \phi, \{b\}, \{c\}, \{d\}, \{e\}, \{a, b\}, \{b, c\}, \{b, d\}, \{b, e\}, \{ c, d\}, \{c, e\}, \{d, e\}, \{a, b, c\}, \{a, b, d\}, \{a, b, e\}, \\\{b, c, d\}, \{b, c, e\}, \{b, d, e\}, \{c, d, e\}, \{a, b, c, d\}, \{a, b, c, e\}, \{a, b, d, e\}, \{b, c, d, e\}\}.
(ii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L}(X) = \{X, \phi, \{a\}, \{b\}, \{c\}, \{e\}, \{a, b\}, \{a, c\}, \{a, e\}, \{b, c\}, \{b, e\}, \{c, e\}, \{d, e\}, \{a, b, c\}, \{a, b, e\}, \\\{a, c, e\}, \{a, d, e\}, \{b, c, e\}, \{b, d, e\}, \{c, d, e\}, \{a, c, d, e\}, \{a, b, c, e\}, \{a, b, d, e\}, \{b, c, d, e\}\}.
(iii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}(X) = P(X).
(iv) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U}(X) = P(X).
(v) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > }(X) = \{X, \phi, \{b\}, \{c\}, \{d\}, \{e\}, \{a, b\}, \{b, c\}, \{b, d, \}, \{b, e\}, \{ c, d\}, \{c, e\}, \{d, e\}, \{a, b, c\}, \{a, b, d\}, \\\{a, b, e\}, \{b, c, d\}, \{b, c, e, \}, \{b, d, e\}, \{ c, d, e\}, \{a, b, c, d\}, \{a, b, c, e\}, \{a, b, d, e\}, \{b, c, d, e\}\}.
(vi) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X) = P(X).
(vii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X) = P(X).
(viii) \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X) = \{X, \phi, \{e\}, \{a, e\}, \{b, e\}, \{c, e\}, \{d, e\}, \{a, b, e\}, \{ a, c, e\}, \{a, d, e\}, \{b, c, e\}, \{b, d, e\}, \{ c, d, e\}, \\\{a, b, c, e\}, \{a, c, d, e\}, \{a, b, d, e\}, \{b, c, d, e\}\}.
It is clear that
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U}(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R}(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R}(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U}(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L}X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L}X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X).
● \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R}(X) is not the dual of \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L} and \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > }(X) is not the dual of \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X).
● In a similar way, I can add examples to show that, \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{L}(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R}(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{I}(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R}(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{U}(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < R > }(X) \nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < U > }(X)\nsubseteq \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X) and \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < L > }(X)\nsubseteq\tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{ < I > }(X).
The aim of this section is to present a new technique to define the approximations of rough sets by using the notion of \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets. Some important significant properties of these approximations are investigated and compared to the previous approximations in Definitions 2.4 [2], 2.7 [8], 2.14 [11] and 2.17 [12]. The techniques in this section and Section 4 are different and independent.
Definition 6.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. The \mathcal{I} - {\bigwedge_{\beta}}_{J} -lower, \mathcal{I} - {\bigwedge_{\beta}}_{J} -upper approximations, \mathcal{I} - {\bigwedge_{\beta}}_{J} -boundary regions and \mathcal{I} - {\bigwedge_{\beta}}_{J} -accuracy of A are defined respectively by:
{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \cup\{G \in \tau^{{\mathcal{I}-{\bigwedge_{\beta}}}}_{J} : G \subseteq A\} = \mathcal{I} - {\bigwedge_{\beta}}_{J} -interior of A .
{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \cap\{H \in \Gamma^{{\mathcal{I}-{\bigwedge_{\beta}}}}_{J}: A \subseteq H\} = \mathcal{I} - {\bigvee_{\beta}}_{J} -closure of A .
{{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)-{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
{{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \frac{|{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)|}{|{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)|}, where |{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)|\neq0.
The following proposition studies the main properties of the current \mathcal{I} - {\bigwedge_{\beta}}_{J} -lower and \mathcal{I} - {\bigwedge_{\beta}}_{J} -upper approximations.
Proposition 6.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A, B \subseteq X. Then,
(i) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) < /italic > < italic > \subseteq A\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) equality hold if A = \phi or X.
(ii) A\subseteq B\Rightarrow {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(iii) A\subseteq B\Rightarrow {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A\cap B)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}}_{J}}(A)\cap{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(v) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}}_{J}}(A\cup B)\supseteq{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\cup{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(vi) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A\cup B) = {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\cup{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(vii) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A\cap B) = {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\cap{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(B).
(viii) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = ({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A^{'}))^{'} , {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = ({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A^{'}))^{'} .
(ix) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)) = {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(x) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)) = {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(xi) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A))\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)).
(xii) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A))\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)).
(xiii) x\in{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\Leftrightarrow G\cap A\neq\phi, \forall G\in \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{J}, x\in G.
(xiv) x\in{\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\Leftrightarrow \exists\; G\in \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{J}, x\in G, G\subseteq A.
The proof of this proposition is simple using \mathcal{I} - {\bigwedge_{\beta}}_{J} -interior and \mathcal{I} - {\bigvee_{\beta}}_{J} -closure, so I omit it.
Remark 6.1. Example 3.3 shows that
(a) The inclusion in Proposition 6.1 parts (i), (iv), (v), (xi) and (xii) can not be replaced by equality relation:
(i) For part (i), if A = \{b, c, e\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = \{a, b, c, e\}, then {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)\nsubseteq A, take A = \{a\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = \phi . Then, A\nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A).
(ii) For part (iv), if A = \{b, c, d, e\}, B = \{a, c, d, e\}, A\cap B = \{c, d, e\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = X, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) = B, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A\cap B) = A\cap B, then {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)\cap {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) = \{a, c, d, e\}\nsubseteq\{c, d, e\} = {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A\cap B).
(iii) For part (v), if A = \{a\}, B = \{b\}, A\cup B = \{a, b\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = \phi, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) = B, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A\cup B) = A\cup B , then {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A\cup B) = \{a, b\} \nsubseteq \{b\} = {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)\cup {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B).
(iv) For part (xi), if A = \{b, c, d, e\}, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) = A, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) = X , then {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) \nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)).
(v) For part (xii), if A = \{a\}, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) = A, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) = \phi , then {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)) \nsubseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}({\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)).
(b) The converse of parts (ii) and (iii) is not necessarily true:
(i) For part (ii), if A = \{a, b, c, e\}, B = \{b, c, d, e\}, then {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = A, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) = X . Therefore, {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) , but A\nsubseteq B.
(ii) For part (iii), if A = \{a\}, B = \{c, d, e\}, then {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) = \phi, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) = B . Therefore, {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(B) , but A\nsubseteq B.
Definition 6.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. A is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -definable ( \mathcal{I} - {\bigwedge_{\beta}}_{J} -exact) set if {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A). Otherwise, A is an \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough set.
In Example 3.3 A = \{c\} is \mathcal{I} - {\bigwedge_{\beta}}_{R} -exact, while B = \{a\} is \mathcal{I} - {\bigwedge_{\beta}}_{R} -rough.
Remark 6.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then the intersection of two \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough sets does not need to be an \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough set as in Example 3.3, \{c, d\} and \{c, e\} , are \mathcal{I} - {\bigwedge_{\beta}}_{R} -rough sets, but \{c, d\}\cap\{c, e\} = \{c\} is not an \mathcal{I} - {\bigwedge_{\beta}}_{R} -rough set.
The following theorem and corollary present the relationships between the current approximations in Definition 6.1 and the previous ones in Definitions 2.4 [2], 2.7 [8] and 2.14 [11].
Theorem 6.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{p}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(ii) {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{s}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(iii) {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\overline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{p}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\alpha}_{J}(A).
(v) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\overline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{s}_{J}(A)\subseteq {\overline{\mathcal{R}}}^{\alpha}_{J}(A).
(vi) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\overline{\mathcal{R}}}_{J}(A).
Proof. (i) By Theorem 2.3 [11], {\underline{\mathcal{R}}}^{\alpha}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{p}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\gamma}_{J}(A)\subseteq{\underline{\mathcal{R}}}^{\beta}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A), and {\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A) = \cup\{G \in \tau^{\bigwedge_{\beta}}_{J} : G \subseteq A:G \subseteq A\}\subseteq \cup\{G \in \tau^{{\mathcal{I}-{\bigwedge_{\beta}}}}_{J} : G \subseteq A\} = {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) (by Proposition 5.1).
(ii) It is similar to (i).
(iii) By Theorem 2.3 [11], {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A), and by (1) {\underline{\mathcal{R}}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A). Hence, {\underline{\mathcal{R}}}_{J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(iv)–(vi) They are similar to (i)–(iii).
Corollary 6.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\mathcal{BND}}^{\bigwedge_{\beta}}_{J}(A)\subseteq {\mathcal{BND}}^{\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\gamma}_{J}(A)\subseteq {\mathcal{BND}}^{p}_{J}(A)\subseteq {\mathcal{BND}}^{\alpha}_{J}(A).
(ii) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\mathcal{BND}}^{\bigwedge_{\beta}}_{J}(A) \subseteq {\mathcal{BND}}^{\beta}_{J}(A)\subseteq {\mathcal{BND}}^{\gamma}_{J}(A)\subseteq {\mathcal{BND}}^{s}_{J}(A)\subseteq {\mathcal{BND}}^{\alpha}_{J}(A) .
(iii) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {\mathcal{BND}}_{J}(A).
(iv) {ACC}^{\alpha}_{J}(A)\leqslant {ACC}^{p}_{J}(A)\leqslant {ACC}^{\gamma}_{J}(A)\leqslant {ACC}^{\beta}_{J}(A)\leqslant{ACC}^{\bigwedge_{\beta}}_{J}(A)\leqslant {ACC}^{\mathcal{I}-{\bigwedge_{\beta}}}_{J}(A).
(v) {ACC}^{\alpha}_{J}(A)\leqslant {ACC}^{s}_{J}(A)\leqslant {ACC}^{\gamma}_{J}(A)\leqslant {ACC}^{\beta}_{J}(A)\leqslant{ACC}^{\bigwedge_{\beta}}_{J}(A)\leqslant {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(vi) {ACC}_{J}(A)\leqslant {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
Corollary 6.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) A is \alpha_J -exact \Rightarrow A is S_J -exact \Rightarrow A is \beta_J -exact \Rightarrow A is {\bigwedge_{\beta}}_{J} -exact \Rightarrow A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -exact.
(ii) A is P_J -exact \Rightarrow A is \beta_J -exact \Rightarrow A is {\bigwedge_{\beta}}_{J} -exact \Rightarrow A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -exact.
(iii) A is {J} -exact \Rightarrow A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -exact.
(iv) A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough \Rightarrow A is {\bigwedge_{\beta}}_{J} -rough \Rightarrow A is {\beta}_{J} -rough \Rightarrow A is S_J -rough \Rightarrow A is \alpha_J -rough.
(v) A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough \Rightarrow A is {\bigwedge_{\beta}}_{J} -rough \Rightarrow A is {\beta}_{J} -rough \Rightarrow A is P_J -rough.
(vi) A is \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough \Rightarrow A is {J} -rough.
The converse of parts of Corollary 6.2 is not necessarily true as in Example 5.2:
(i) If A = \{a\}, then it is \mathcal{I} - {\bigwedge_{\beta}}_{R} -exact, but it is neither {\bigwedge_{\beta}}_{R} -exact nor {R} -exact.
(ii) If A = \{b\}, then it is {R} -rough and {\bigwedge_{\beta}}_{R} -rough, but it is not \mathcal{I} - \beta_{R} -rough.
The following proposition and corollary are introduced the relationships between the current approximations in Definition 6.1 and the previous one in Definition 2.17 [12].
Proposition 6.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {\underline{\mathcal{R}}^{\mathcal{I}-P}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(ii) {\underline{\mathcal{R}}^{\mathcal{I}-\alpha}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-S}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) .
(iii) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-P}_J}(A).
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq{\overline{\mathcal{R}}^{\mathcal{I}-\beta}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-S}_J}(A)\subseteq {\overline{\mathcal{R}}^{\mathcal{I}-\alpha}_J}(A).
Proof. By Proposition 5.2, the proof is obvious.
Corollary 6.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\beta}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-P}_J}(A).
(ii) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\beta}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-S}_J}(A)\subseteq {{\mathcal{BND}}^{\mathcal{I}-\alpha}_J}(A).
(iii) {{ACC}^{\mathcal{I}-P}_J}(A)\leqslant {{ACC}^{\mathcal{I}-\beta}_J}(A)\leqslant {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) .
(iv) {{ACC}^{\mathcal{I}-\alpha}_J}(A)\leqslant {{ACC}^{\mathcal{I}-S}_J}(A)\leqslant {{ACC}^{\mathcal{I}-\beta}_J}(A)\leqslant {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
Corollary 6.4. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) A is J -exact \Rightarrow A is \mathcal{I} - \alpha_J -exact \Rightarrow A is \mathcal{I} - S_J -exact \Rightarrow A is \mathcal{I} - \beta_J -exact \Rightarrow A is \mathcal{I} - {\bigwedge_{\beta}}_J -exact.
(ii) A is \mathcal{I} - P_J -exact \Rightarrow A is \mathcal{I} - \beta_J -exact \Rightarrow A is \mathcal{I} - {\bigwedge_{\beta}}_J -exact.
(iii) A is \mathcal{I} - {\bigwedge_{\beta}}_J -rough \Rightarrow A is \mathcal{I} - {\beta}_{J} -rough \Rightarrow A is \mathcal{I} - S_J -rough \Rightarrow A is \mathcal{I} - \alpha_J -rough.
(iv) A is \mathcal{I} - {\bigwedge_{\beta}}_J -rough \Rightarrow A is \mathcal{I} - {\beta}_{J} -rough \Rightarrow A is \mathcal{I} - P_J -rough.
Remark 6.3. Example 3.3 shows that the converse of the implications in Corollaries 6.3, 6.4 and Proposition 6.2 is not true in general.
In Table 2, the lower, upper approximations, boundary region and accuracy are calculated by using Hosny's method 2.17 [12] and the current approximations in Definition 6.1 by using Example 3.3.
A | Hosny's method 17 [12] | The current method in Definition 6.1 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | |
\{a\} | \phi | \{a\} | \phi | 0 | \phi | \{a\} | \{a\} | 0 |
\{b\} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} |
\{c\} | \{c\} | \{c\} | \phi | 1 | \{c\} | \{c\} | \phi | 1 |
\{d\} | \{d\} | \{d\} | \phi | 1 | \{d\} | \{d\} | \phi | 1 |
\{e\} | \phi | \{e\} | \{e\} | 0 | \{e\} | \{e\} | \phi | 1 |
\{a, b\} | \{a, b\} | \{a, b\} | \phi | 1 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} |
\{a, d\} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} |
\{a, e\} | \phi | \{a, e\} | \{a, e\} | 0 | \{e\} | \{a, e\} | \{a\} | \frac{1}{2} |
\{b, c\} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} |
\{b, d\} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} |
\{b, e\} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} |
\{c, d\} | \{c, d\} | \{c, d\} | \phi | 1 | \{c, d\} | \{c, d\} | \phi | 1 |
\{c, e\} | \{c, e\} | \{c, e\} | \phi | 1 | \{c, e\} | \{c, e\} | \phi | 1 |
\{d, e\} | \{d, e\} | \{d, e\} | \phi | 1 | \{d, e\} | \{d, e\} | \phi | 1 |
\{a, b, c\} | \{a, b, c\} | \{a, b, c\} | \phi | 1 | \{a, b, c\} | \{a, b, c\} | \phi | 1 |
\{a, b, d\} | \{a, b, d\} | \{a, b, d\} | \phi | 1 | \{a, b, d\} | \{a, b, d\} | \phi | 1 |
\{a, b, e\} | \{a, b, e\} | \{a, b, e\} | \phi | 1 | \{a, b, e\} | \{a, b, e\} | \phi | 1 |
\{a, c, d\} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} |
\{a, c, e\} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} |
\{a, d, e\} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} |
\{b, c, d\} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} |
\{b, c, e\} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} |
\{b, d, e\} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} |
\{c, d, e\} | \{c, d, e\} | \{c, d, e\} | \phi | 1 | \{c, d, e\} | \{c, d, e\} | \phi | 1 |
\{a, b, c, d\} | \{a, b, c, d\} | X | \{e\} | \frac{4}{5} | \{a, b, c, d\} | \{a, b, c, d\} | \phi | 1 |
\{a, b, c, e\} | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 |
\{a, b, d, e\} | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 |
\{a, c, d, e\} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} |
\{b, c, d, e\} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} |
X | X | X | \phi | 1 | X | X | \phi | 1 |
For example, take A = \{e\}, then the boundary and accuracy by the present method in Definition 6.1 are \phi and 1 respectively. Whereas, the boundary and accuracy by using Hosny's method 2.17 [12] are \{e\} and 0 respectively.
Remark 6.4. It should be noted that the \mathcal{I} - {\bigwedge_{\beta}}_{J} -approximations in this section and the \mathcal{I} - {\delta\beta}_{J} -approximations in Section 4 are different and independent. As, the concepts of \mathcal{I} - {\delta\beta}_{J} -open sets and \mathcal{I} - {\bigwedge_{\beta}}_{J} -sets are different and independent as shown in Remark 5.5.
This section concentrates on generalization the concept of rough membership functions by introducing the concepts of \mathcal{I} - {\delta\beta}_{J} -rough membership functions and \mathcal{I} - {\bigwedge_{\beta}}_{J} -rough membership functions.
Definition 7.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X, x\in X and A \subseteq X:
(i) If x\in {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A), then x is J - {\delta\beta} -surely with respect to \mathcal{I} ( \mathcal{I}-{\delta\beta}_J -surely) belongs to A , denoted by x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(ii) If x\in{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A), then x is J - {\delta\beta} -possibly with respect to \mathcal{I} (briefly \mathcal{I}-{\delta\beta}_J -possibly) belongs to A , denoted by x\; {\overline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(iii) If x\in {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A), then x is J - {\bigwedge_{\beta}} -surely with respect to \mathcal{I} ( \mathcal{I}-{\bigwedge_{\beta}}_J -surely) belongs to A , denoted by x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(iv) If x\in{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A), then x is J - {\bigwedge_{\beta}} -possibly with respect to \mathcal{I} (briefly \mathcal{I}-{\bigwedge_{\beta}}_J -possibly) belongs to A , denoted by x\; {\overline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A.
It is called J - {\delta\beta} -strong ( J - {\bigwedge_{\beta}} -strong) and J - {\delta\beta} -weak ( J - {\bigwedge_{\beta}} -weak) membership relations with respect to \mathcal{I} respectively.
Remark 7.1. According to Definitions 4.1 and 6.1, the \mathcal{I} - {\delta\beta}_{J} -lower, \mathcal{I} - {\delta\beta}_{J} -upper approximations, \mathcal{I} - {\bigwedge_{\beta}}_{J} -lower and \mathcal{I} - {\bigwedge_{\beta}}_{J} -upper approximations for any A\subseteq X can be written as:
(i) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) = \{x\in X: x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A\}.
(ii) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) = \{x\in X: x\; {\overline{\in}}^{\mathcal{I}-{\delta\beta}_J} A\}.
(iii) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \{x\in X: x\; {\underline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A\}.
(iv) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \{x\in X: x\; {\overline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A\}.
Lemma 7.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) If x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A, then x \in A.
(ii) If x \in A, then x\; {\overline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(iii) If x\; {\underline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A, then x \in A.
(iv) If x \in A, then x\; {\overline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A.
Proof. Straightforward.
Remark 7.2. The converse of Lemma 7.1 is not true in general, as it is shown in Example 3.3 that if:
(i) A = \{a, b, c, d\}, then a \in A, but a{\underline{\not\in}}^{\mathcal{I}-{\delta\beta}_R}A.
(ii) A = \{a, b, c, d\}, then e\; {\overline{\in}}^{\mathcal{I}-{\delta\beta}_R} A, but e \not\in A.
(iii) A = \{a, c, d, e\}, then a \in A, but a\; {\underline{\not\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_R} A.
(iv) A = \{b, c, d\}, then e\; {\overline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_R} A, but e \not\in A.
Proposition 7.1. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) If x\; {\underline{\in}}_{J} A\Rightarrow x\; {\underline{\in}}^{{\eta}}_{J} A\Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\eta}}_{J} A\Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(ii) If x\; {\overline{\in}}^{\mathcal{I}-{\delta\beta}_J} A\Rightarrow x\; {\overline{\in}}^{\mathcal{I}-{\eta}}_{J} A\Rightarrow x \; {\overline{\in}}^{{\eta}}_{J} A\Rightarrow x \; {\overline{\in}}_{J} A.
(iii) If x\; {\underline{\in}}_{J} A\Rightarrow x\; {\underline{\in}}^{{\eta}}_{J} A\Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\eta}}_{J} A\Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A.
(iv) If x\; {\overline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A\Rightarrow x\; {\overline{\in}}^{\mathcal{I}-{\eta}}_{J} A\Rightarrow x \; {\overline{\in}}^{{\eta}}_{J} A\Rightarrow x \; {\overline{\in}}_{J} A.
Proof. I prove (i) and the other similarly. x\; {\underline{\in}}_{J} A\Rightarrow x\; {\underline{\in}}^{{\eta}}_{J} A\Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\eta}}_{J} A by Proposition 2.6. Let x\; {\underline{\in}}^{\mathcal{I}-{\eta}}_{J} A. Then, x\in{\underline{\mathcal{R}}^{\mathcal{I}-\eta}_{J}}(A)\Rightarrow x\in {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) (\; by\; Theorem\; 4.1) \Rightarrow x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
Remark 7.3. The converse of Proposition 7.1 is not true in general, as it is shown in
(i) Example 3.2 that if A = \{b, c\}, then b\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_R} A, but b\; {\underline{\not\in}}^{{\beta}}_{R} A.
(ii) Example 3.2 that if A = \{b, c\}, then a\; {\overline{\in}}^{\mathcal{I}-{\beta}}_{R} A, but a\; {\overline{\not\in}}^{\mathcal{I}-{\delta\beta}_R} A.
(iii) Example 3.3 that if A = \{e\}, then e\; {\underline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_R} A, but e\; {\underline{\not\in}}^{{\beta}}_{R} A.
(iv) Example 3.3 that if A = \{a, b, c, d\}, then e\; {\overline{\in}}^{\mathcal{I}-{\beta}}_{R} A, but e\; {\overline{\not\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_R}.
Definition 7.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X, A \subseteq X and x \in X. The {\mathcal{I}-{\delta\beta}_J} -rough membership functions of A are defined by \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}\rightarrow [0, 1], where
\mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = \{^{1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;if \; 1\in\psi^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x).}_{min(\psi^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x))\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise.}\}. \\{\rm{and}}\; \psi^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = \frac{|\mathcal{I}-{{\delta\beta}_J}(x)\cap A|}{|\mathcal{I}-{{\delta\beta}_J}(x)|}, x\in\mathcal{I}-{{\delta\beta}_J}(x), \mathcal{I}-{{\delta\beta}_J}(x) \in \mathcal{I}-{\delta\beta}_{J}O(X). |
Remark 7.4. The {\mathcal{I}-{\delta\beta}_J} -rough membership functions are used to define the \mathcal{I} - {\delta\beta}_{J} -lower and \mathcal{I} - {\delta\beta}_{J} -upper approximations as follows:
(i) {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) = \{x\in X:\mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 1\}.
(ii) {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) = \{x\in X:\mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) > 0\}.
(iii) {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}}(A) = \{x\in X:0 < \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) < 1\}.
The following results give the fundamental properties of the {\mathcal{I}-{\delta\beta}_J} -rough membership functions.
Proposition 7.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A, B \subseteq X. Then
(i) If \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 1\Leftrightarrow x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A.
(ii) If \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 0\Leftrightarrow x\in X-{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A).
(iii) If 0 < \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) < 1\Leftrightarrow x\in {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{J}}(A).
(iv) If \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A^{'}}(x) = 1-\mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x), \forall\; x\in X.
(v) If \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A \cup B}(x)\geq max(\mu^{\mathcal{I}-{{\zeta}_J}}_{A}(x), \mu^{{\mathcal{I}-{\delta\beta}_J}}_{B}(x)), \forall\; x\in X.
(vi) If \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A \cap B}(x)\leq min(\mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x), \mu^{{\mathcal{I}-{\delta\beta}_J}}_{B}(x)), \forall\; x\in X.
Proof. I prove (i), and the others similarly.
x\; {\underline{\in}}^{\mathcal{I}-{\delta\beta}_J} A \Leftrightarrow x\in {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A). Since {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) is \mathcal{I}-{\delta\beta}_{J} -open set contained in A , thus \frac{|{\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A) \cap A|}{| {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)(A)|} = \frac{| {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)|}{| {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)|} = 1. Then, 1\in \psi^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) and accordingly \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 1.
The following lemma is very interesting since it is given the relations between the J -rough membership relations [3], J -nearly rough membership relations [3], J -nearly rough membership relations with respect to \mathcal{I} [12] and \mathcal{I} - {\delta\beta}_{J} -rough membership functions.
Lemma 7.2. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) \mu^{J}_{A}(x) = 1\Rightarrow \mu^{{{\eta}_J}}_{A}(x) = 1\Rightarrow \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 1\Rightarrow \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 1, \forall\; x\in X.
(ii) \mu^{J}_{A}(x) = 0\Rightarrow \mu^{{\eta_{J}}}_{A}(x) = 0\Rightarrow \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 0\Rightarrow \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 0, \forall\; x\in X.
Proof. (i) \mu^{J}_{A}(x) = 1\Rightarrow \mu^{{{\eta}_J}}_{A}(x) = 1\Rightarrow \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 1 directly from Lemma 2.1. Let \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 1, then x\in{\underline{\mathcal{R}}^{\mathcal{I}-\eta}_{J}}(A)\Rightarrow x\in {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)\Rightarrow \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 1, \forall\; x\in X.
(ii) \mu^{J}_{A}(x) = 0\Rightarrow \mu^{{\eta_{J}}}_{A}(x) = 0 directly from Lemma 2.1. Let \mu^{{\eta_{J}}}_{A}(x) = 0, then x\in X-{\overline{\mathcal{R}}^{\mathcal{I}-\eta}_{J}}(A)\Rightarrow x\in X-{\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_J}(A)\Rightarrow \mu^{{\mathcal{I}-{\delta\beta}_J}}_{A}(x) = 0, \forall\; x\in X.
Remark 7.5. The converse of Lemma 7.2 is not true in general, as it is shown in Example 3.2.
Definition 7.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X, A \subseteq X and x \in X. The {\mathcal{I}-{\bigwedge_{\beta}}_J} -rough membership functions of a J -ndS on X for a A are defines by \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}\rightarrow [0, 1], where
\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = \{^{1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;if \; 1\in\psi^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x).}_{min(\psi^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x))\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise.}\}.\\ {\rm{and}}\; \psi^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = \frac{|\mathcal{I}-{{\bigwedge_{\beta}}_J}(x)\cap A|}{|\mathcal{I}-{{\bigwedge_{\beta}}_J}(x)|}, x\in\mathcal{I}-{{\bigwedge_{\beta}}_J}(x), \mathcal{I}-{{\bigwedge_{\beta}}_J}(x) \in \mathcal{I}-{\bigwedge_{\beta}}_{J}O(X). |
Remark 7.6. The \mathcal{I} - J -nearly rough membership functions are used to define the \mathcal{I} - {\bigwedge_{\beta}}_{J} -lower and \mathcal{I} - {\bigwedge_{\beta}}_{J} -upper approximations as follows:
(i) {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \{x\in X:\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = 1\}.
(ii) {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \{x\in X:\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) > 0\}.
(iii) {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A) = \{x\in X:0 < \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) < 1\}.
The following results give the fundamental properties of the {\mathcal{I}-{\bigwedge_{\beta}}_J} -rough membership functions.
Proposition 7.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A, B \subseteq X. Then
(i) If \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = 1\Leftrightarrow x {\underline{\in}}^{\mathcal{I}-{\bigwedge_{\beta}}_J} A.
(ii) If \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = 0\Leftrightarrow x\in X-{\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(iii) If 0 < \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) < 1\Leftrightarrow x\in {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{J}}}(A).
(iv) If \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A^{'}}(x) = 1-\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x), \forall\; x\in X.
(v) If \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A \cup B}(x)\geq max(\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x), \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{B}(x)), \forall\; x\in X.
(vi) If \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A \cap B}(x)\leq min(\mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x), \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{B}(x)), \forall\; x\in X.
Proof. It is similar to Proposition 7.2.
Lemma 7.3. Let (X, \mathcal{R}, \Xi_J) be a J -ndS, \mathcal{I} be an ideal on X and A \subseteq X. Then
(i) \mu^{J}_{A}(x) = 1\Rightarrow \mu^{{{\eta}_J}}_{A}(x) = 1\Rightarrow \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 1 \Rightarrow \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = 1, \forall\; x\in X.
(ii) \mu^{J}_{A}(x) = 0\Rightarrow \mu^{{\eta_{J}}}_{A}(x) = 0\Rightarrow \mu^{\mathcal{I}-{\eta_{J}}}_{A}(x) = 0\Rightarrow \mu^{{\mathcal{I}-{\bigwedge_{\beta}}_J}}_{A}(x) = 0, \forall\; x\in X.
Proof. It isimilar to Lemma 7.2.
Remark 7.7. The converse of Lemma 7.3 is not true in general, as it is shown in Example 3.3.
Remark 7.8. According to Lemmas 7.2 and 7.3, the current Definitions 7.2 and 7.3 are also generalization of the approaches in [22] and 2.19 [28].
Finally in this section, an applied example in Chemistry field is introduced by applying the present Definition 3.1 and the previous one 2.6 in [8]. Furthermore, a practical example uses an equivalence relation that induced from an information system is introduced to compare between the current approach in Definition 7.2 and the previous approach for Pawlak and Skoworn 2.19 [28].
Example 8.1. Let X = \{x_1, x_2, x_3, x_4, x_5\} be five amino acids (for short, AAs). The (AAs) are described in terms of seven attributes: a_1 = PIE and a_2 = SAC = surface area, a_3 = MR = molecular refractivity, a_4 = LAM = the side chain polarity and a_5 = Vol = molecular volume ([10,34]). Table 3 shows all quantitative attributes of five AAs.
a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | |
\{x_{1}\} | 0.23 | 254.2 | 2.216 | -0.02 | 82.2 |
\{x_{2}\} | -0.48 | 303.6 | 2.994 | -1.24 | 112.3 |
\{x_{3}\} | -0.61 | 287.9 | 2.994 | -1.08 | 103.7 |
\{x_{4}\} | 0.45 | 282.9 | 2.933 | -0.11 | 99.1 |
\{x_{5}\} | -0.11 | 335.0 | 3.458 | -0.19 | 127.5 |
I consider the relations on X defined as: \mathcal{R}_i = \{(x_i, x_j) : x_i(a_k)-x_j(a_k) < \frac{\sigma_{k}}{2}, i, j, k = 1, 2, ..., 5\} where \sigma_{k} represents the standard deviation of the quantitative attributes.
The right neighborhoods \forall x\in X with respect to the relations are shown in Table 4.
$$ | x_i\mathcal{R}_{1} | x_i\mathcal{R}_{2} | x_i\mathcal{R}_{3} | x_i\mathcal{R}_{4} | x_i\mathcal{R}_{5} |
\{x_{1}\} | \{x_1, x_4\} | X | X | \{x_1, x_4, x_5\} | X |
\{x_{2}\} | X | \{x_2, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_5\} |
\{x_{3}\} | X | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_3, x_4, x_5\} |
\{x_{4}\} | \{x_4\} | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | \{x_1, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} |
\{x_{5}\} | \{x_1, x_4, x_5\} | \{x_5\} | \{x_5\} | \{x_1, x_4, x_5\} | \{x_3, x_5\} |
The intersection of all right neighborhoods \forall x\in X is:
x_1\mathcal{R} = {\cap}_{k = 1}^{5}(x_1\mathcal{R}_k) = \{x_1, x_4\}, x_2\mathcal{R} = {\cap}_{k = 1}^{5}(x_2\mathcal{R}_k) = \{x_2, x_5\}, x_3\mathcal{R} = {\cap}_{k = 1}^{5}(x_3\mathcal{R}_k) = \{x_2, x_3, x_4, x_5\}, x_4\mathcal{R} = {\cap}_{k = 1}^{5}(x_4\mathcal{R}_k) = \{x_4\}, x_5\mathcal{R} = {\cap}_{k = 1}^{5}(x_5\mathcal{R}_k) = \{x_5\}. Then, \tau_{R} = \{\phi, X, \{x_4\}, \{x_5\}, \{x_4, x_5\}, \{x_1, x_4\}, \{x_2, x_5\}, \{x_1, x_4, x_5\}, \{x_2, x_4, x_5\}, \{x_1, x_2, x_4, x_5\}, \{x_1, x_3, x_4, x_5\}\}, {\beta}_{R}O(X) = \{\phi, X, \{x_4\}, \{x_5\}, \{x_1, x_4\}, \{x_1, x_5\}, \{x_2, x_5\}, \{x_3, x_4\}, \{x_3, x_5\}, \{x_4, x_5\}, \{x_1, x_3, x_5\}, \{x_1, x_4, x_5\}, \{x_2, x_3, x_5\}, \{x_2, x_4, x_5\}, ,\{x_3, x_4, x_5\}, \{x_1, x_2, x_4, x_5\}, \{x_1, x_3, x_4, x_5\}, \{x_2, x_3, x_4, x_5\}\}. Let \mathcal{I} = \{\phi, \{x_1\}\}, then \mathcal{I} - {\delta\beta}_{R}O(X) = P(X).
(i) It is clear that every \beta_R -open is \mathcal{I} - {\delta\beta}_{R} -open, but the converse is not necessary to be true. For example take A = \{x_1\} which is \mathcal{I} - {\delta\beta}_{R} -open, but it is not \beta_R -open. Hence, the current concept generalize and extend the previous one 2.6 in [8].
(ii) The current approximations which are depended on \mathcal{I} - {\delta\beta}_{R} -open is better than the previous approximations 2.7 [8] which depended on {\beta}_{R} -open. As for any concept A\subseteq X (collection of Amino Acid), this concept is determine by the lower and upper approximations which defines its boundary. Moreover, the accuracy increases by the decreases of the boundary region. Clearly the accuracy measure by using the suggested class \mathcal{I} - {\delta\beta}_{R} -open in general is greater than the accuracy measure by using \beta_R -open. For example take A = \{x_1, x_4, x_5\}, Then,
(a) by the current Definition 4.1, {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = \phi and {{ACC}^{\mathcal{I}-{\delta\beta}}_{R}}(A) = 1 ;
(b) by the previous one in Definition 2.7 [8], {\mathcal{BND}}^{\beta}_R(A) = \{2, 3\} and {ACC}^{\beta}_R(A) = \frac{3}{5}.
(iii) Similarly, it is easy to calculate \mathcal{I} - {\beta}_{R}O(X), \tau^{\mathcal{I}-{\bigwedge_{\beta}}}_{R} and their approximations by the same manner in Tables 1 and 2. This also shows that the present methods is better than the previous ones in [2,8,11,12].
Example 8.2. Consider the following information system as in Table 5. The data about six students is given as shown below:
Student | Science | German | Mathematics | Decision |
\{x_{1}\} | Bad | Good | Medium | Accept |
\{x_{2}\} | Good | Bad | Medium | Accept |
\{x_{3}\} | Good | Good | Good | Accept |
\{x_{4}\} | Bad | Good | Bad | Reject |
\{x_{5}\} | Good | Bad | Medium | Reject |
\{x_{6}\} | Bad | Good | Good | Accept |
From Table 5:
(i) The set of universe: X = \{x_1, x_2, x_3, x_4, x_5, x_6\}.
(ii) The set of attributes: AT = \{Science, \; German, \; Mathematics\}.
(iii) The sets of values:
\begin{align*} V_{Science} & = \{Bad, Good\},\\ V_{German} & = \{Bad, Good\},\\ V_{Mathematics} & = \{Bad, Medium, Good\} \end{align*} |
and
\begin{align*} V_{Decision} & = \{Accept, Reject\}. \end{align*} |
I take the set of condition attributes, C = \{Science, German, Mathematics\}. Thus, the corresponding equivalence relation is \mathcal{R} = \{(x_1, x_1), (x_2, x_5), (x_3, x_3), (x_4, x_4), (x_5, x_2), (x_6, x_6)\}, let \mathcal{I} = \{\phi, \{x_1\}\}. Then, {\mathcal{I}} - {\delta\beta}_{R}O(X) = P(X). Let A (Decision: Accept) = \{x_1, x_2, x_3, x_6\}. Then
(i) The rough membership functions with respect to the Definition of Pawlak and Skowron 19 [28] are computed as follows:
\begin{align*} \mu^{A}(x_1) = \mu^{A}(x_3) = \mu^{A}(x_6) = 1, \mu^{A}(x_2) = \frac{1}{2}. \end{align*} |
(ii) The \mathcal{I} - {\delta\beta}_{R} -rough membership functions in Definition 7.2 are calculated as follows:
\begin{align*} \mu^{{\mathcal{I}-{\delta\beta}_{R}}}_{A}(x_1) = \mu^{{\mathcal{I}-{\delta\beta}_{R}}}_{A}(x_2) = \mu^{{\mathcal{I}-{\delta\beta}_{R}}}_{A}(x_3) = \mu^{{\mathcal{I}-{\delta\beta}_{R}}}_{A}(x_6) = 1. \end{align*} |
Obviously, the current Definition 7.2 is accurate more than the Definition of Pawlak and Skowron 2.19 [28].
Remark 8.1. It should be noted that for some elements that have decision (Reject) such that x_5
(i) The rough membership function with respect to the Definition of Pawlak and Skowron 2.19 [28] is \mu^{A}(x_5) = \frac{1}{2}. This means that x_5 may belong to the set A (Decision: Accept), A = \{x_1, x_2, x_3, x_6\} and this contradicts to Table 5.
(ii) The \mathcal{I} - {\delta\beta}_{R} -rough membership function in Definition 7.2 is \mu^{{\mathcal{I}-{\delta\beta}_{R}}}_{A}(x_5) = 0. This means that x_{5}\not\in A (Decision: Accept) = \{x_1, x_2, x_3, x_6\} which is coincide with Table 5.
Rough set theory is a vast area that has varied inventions, applications and interactions with many other branches of mathematical sciences. Deriving rough sets from topology is one such interaction. There is a close homogeneity between rough set theory and general topology. Topology is a rich source for constructs that can be helpful to enrich the original model of approximation spaces. Ideal is a fundamental concept in topological spaces and played an important role in the study of a generalization of rough set. Since the advent of the ideals, several research papers with interesting results in different respects came to existence. In the current results, ideals were very helpful for increasing the current lower approximations and decreasing the current upper approximations. Consequently, they reduced the boundary region and increased the accuracy measure. So, they removed the vagueness of a concept that is an essential goal for the rough set. The properties of the proposed concepts and methods were studied. It should be noted that the two methods in this paper were different and independent as it was shown. I gave not only their characterizations but also discussed the relationships among them and between the previous ones and shown to be more general. The present accuracy measures were more accurate and higher than the previous ones. Since, the boundary regions were decreased (or empty) by increasing the lower approximations and decreasing the upper approximations. Further, two kind of the rough membership functions with respect to ideals were introduced as extension of the former functions. Moreover, an applied example in chemical field was suggested by applying the current methods to illustrate the concepts in a friendly way. Finally, a particle example was provided to clarify the technique of the present rough membership functions and demonstrate their utility and efficiency. I hope the beauty of this work can pave way to many other research fields such as:
(i) Fuzzy topologies, soft topologies and Multiset topologies.
(ii) New applications of these new approximations in various real-life fields.
This is a part of the future research.
The author extend her appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P.1/15/42). She also would like to express her sincere thanks to the editor and anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper.
This work does not have any conflicts of interest.
[1] | M. E. Abd El-Monsef, S. N. El-Deeb, R. A. Mahmoud, \beta-open sets and \beta-continuous mappings, Bull. Fac. Sci. Assiut. Univ., 12 (1983), 77–90. |
[2] | M. E. Abd El-Monsef, O. A. Embaby, M. K. El-Bably, Comparison between rough set approximations based on different topologies, Int. J. Granular Comput., Rough Sets Intell. Syst., 3 (2014), 292–305. |
[3] | M. E. Abd El-Monsef, A. M. Kozae, M. K. El-Bably, New generalized definitions of rough membership relations and functions from topological point of view, J. Adv. Math., 8 (2014), 1635–1652. |
[4] |
H. M. Abo-Doniaa, New rough set approximation spaces, Abstr. Appl. Anal., 2013, (2013). doi: 10.1155/2013/189208. doi: 10.1155/2013/189208
![]() |
[5] |
H. M. Abo-Doniaa, A. S. Salama, Generalization of Pawlak's rough approximation spaces by using \delta\beta-open sets, Int. J. Approx. Reason., 53 (2012), 1094–1105. doi: 10.1016/j.ijar.2012.05.001. doi: 10.1016/j.ijar.2012.05.001
![]() |
[6] | H. M. Abo-Doniaa, A. S. Salama, \beta-approximation spaces, J. Hybrid Comput. Res., 1 (2008). |
[7] |
T. M. Al-shami, H. Işık, A. S. Nawar, R. A. Hosny, Some topological approaches for generalized rough sets via ideals, Math. Probl. Eng., 2021 (2021). doi: 10.1155/2021/5642982. doi: 10.1155/2021/5642982
![]() |
[8] |
W. S. Amer, M. I. Abbas, M. K. El-Bably, On J-nearly concepts in rough sets with some applications, Int. J. Fuzzy Intell. Syst., 32 (2017), 1089–1099. doi: 10.3233/JIFS-16169. doi: 10.3233/JIFS-16169
![]() |
[9] |
L. D'eera, C. Cornelis, A comprehensive study of fuzzy covering-based rough set models: Definitions, properties and interrelationships, Fuzzy Sets Syst., 336 (2018), 1–26. doi: 10.1016/j.fss.2017.06.010. doi: 10.1016/j.fss.2017.06.010
![]() |
[10] |
N. E. El-Tayar, R. S. Tsai, P. A. Carruptand, B. Testa, Octan-1-ol-water partition coefficients of zwitterionic \alpha-amino acids. Determination by centrifugal partition chromatography and factorization into steric/hydrophobic and polar components, J. Chem. Soc., 2 (1992), 79–84. doi: 10.1039/P29920000079. doi: 10.1039/P29920000079
![]() |
[11] |
M. Hosny, On generalization of rough sets by using two different methods, J. Intell. Fuzzy Syst., 35 (2018), 979–993. doi: 10.3233/JIFS-172078. doi: 10.3233/JIFS-172078
![]() |
[12] |
M. Hosny, Topological approach for rough sets by using J-nearly concepts via ideals, Filomat, 34 (2020), 273–286. doi: 10.2298/FIL2002273H. doi: 10.2298/FIL2002273H
![]() |
[13] |
M. Hosny, Idealization of j-approximation spaces, Filomat, 34 (2020), 287–301. doi: 10.2298/FIL2002287H. doi: 10.2298/FIL2002287H
![]() |
[14] |
R. A. Hosny, B. A. Asaad, A. A. Azzam, T. M. Al-Shami, Various topologies generated from E_j-neighbourhoods via ideals, Complexity, 2021 (2021). doi: 10.1155/2021/4149368. doi: 10.1155/2021/4149368
![]() |
[15] |
Z. Huang, J. Li, A fitting model for attribute reduction with fuzzy \beta covering, Fuzzy Sets Syst., 413 (2021), 114–137. doi: 10.1016/j.fss.2020.07.010. doi: 10.1016/j.fss.2020.07.010
![]() |
[16] |
Z. Huang, J. Li, Y. Qian, Noise-tolerant fuzzy \beta covering based multigranulation rough sets and feature subset selection, IEEE Trans. Fuzzy Syst., (2021), 1–15. doi: 10.1109/TFUZZ.2021.3093202. doi: 10.1109/TFUZZ.2021.3093202
![]() |
[17] |
D. Jankovic, T. R. Hamlet, New topologies from old via ideals, Am. Math. Mon., 97 (1990), 295–310. doi: 10.1080/00029890.1990.11995593. doi: 10.1080/00029890.1990.11995593
![]() |
[18] | K. Kuratowski, Topology, Vol. I, New York: Academic Press, 1966. |
[19] |
N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Mon., 70 (1963), 36–41. doi: 10.1080/00029890.1963.11990039. doi: 10.1080/00029890.1963.11990039
![]() |
[20] |
X. Li, S. Liu, Matroidal approaches to rough sets via closure operators, Int. J. Approx. Reason., 53 (2012), 513–527. doi: 10.1016/j.ijar.2011.12.005. doi: 10.1016/j.ijar.2011.12.005
![]() |
[21] |
Z. Li, T. Xie, Q. Li, Topological structure of generalized rough sets, Comput. Math. Appl., 63 (2012), 1066–1071. doi: 10.1016/j.camwa.2011.12.011. doi: 10.1016/j.camwa.2011.12.011
![]() |
[22] | T. Y. Lin, Granular computing on binary relation I: Data mining and neighborhood systems, In: L. Polkowski, A. Skowron, Rough Sets in knowledge discovery 1, Heidelberg: Physica-Verlag, (1998), 107–121. |
[23] |
X. Ma, Q. Liu, J. Zhan, A survey of decision making methods based on certain hybrid soft set models, Artif. Intell. Rev., 47 (2017), 507–530. doi: 10.1007/s10462-016-9490-x. doi: 10.1007/s10462-016-9490-x
![]() |
[24] | A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, On pre continuous and weak pre continuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53. |
[25] |
O. Njestad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970. doi: 10.2140/pjm.1965.15.961
![]() |
[26] |
Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci., 11 (1982), 341–356. doi: 10.1007/BF01001956
![]() |
[27] | Z. Pawlak, Rough concept analysis, Bull. Pol. Acad. Sci. Math., 33 (1985), 495–498. |
[28] | Z. Pawlak, A. Skowron, Rough membership function, In: R. E. Yeager, M. Fedrizzi, J. Kacprzyk, Advaces in the Dempster-Schafer of evidence, New York: Wiley, (1994), 251–271. |
[29] |
Z. Pei, D. Pei, Li. Zheng, Topology vs generalized rough sets, Int. J. Approx. Reason., 52 (2011), 231–239. doi: 10.1016/j.ijar.2010.07.010. doi: 10.1016/j.ijar.2010.07.010
![]() |
[30] |
B. Roy, On weakly (\mu, \lambda)-open functions, Ukrainian Math. J., 66 (2015), 1595–1602. doi: 10.1007/s11253-015-1035-y
![]() |
[31] | A. S. Salama, Some topological properties of rough sets with tools for data mining, Int. J. Comput. Sci., 8 (2011), 588–595. |
[32] |
A. S. Salama, M. M. E. Abd El-Monsef, Generalizations of rough set concepts, J. King Saud Univ.-Sci., 23 (2011), 17–21. doi: 10.1016/j.jksus.2010.06.001. doi: 10.1016/j.jksus.2010.06.001
![]() |
[33] |
R. Vaidynathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20 (1944), 51–61. doi: 10.1007/BF03048958. doi: 10.1007/BF03048958
![]() |
[34] |
B. Walczak, D. L. Massart, Tutorial Rough sets theory, Chemometr. Intell. Lab. Syst., 47 (1999), 1–16. doi: 10.1016/S0169-7439(98)00200-7. doi: 10.1016/S0169-7439(98)00200-7
![]() |
1. | Tareq M. Al-Shami, Mona Hosny, Improvement of Approximation Spaces Using Maximal Left Neighborhoods and Ideals, 2022, 10, 2169-3536, 79379, 10.1109/ACCESS.2022.3194562 | |
2. | Mona Hosny, Tareq M. Al-shami, Rough set models in a more general manner with applications, 2022, 7, 2473-6988, 18971, 10.3934/math.20221044 | |
3. | M. Hosny, Tareq M. Al-shami, Abdelwaheb Mhemdi, Novel approaches of generalized rough approximation spaces inspired by maximal neighbourhoods and ideals, 2023, 69, 11100168, 497, 10.1016/j.aej.2023.02.008 | |
4. | Mona Hosny, Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications, 2022, 7, 2473-6988, 13104, 10.3934/math.2022724 | |
5. | Nurettin Bağırmaz, A topological approach for rough semigroups, 2024, 9, 2473-6988, 29633, 10.3934/math.20241435 | |
6. | Murad ÖZKOÇ, Büşra KÖSTEL, On the topology $ \tau^{\diamond}_R $ of primal topological spaces, 2024, 9, 2473-6988, 17171, 10.3934/math.2024834 | |
7. | Heba Mustafa, Tareq Al-Shami, Ramy Wassef, Rough set paradigms via containment neighborhoods and ideals, 2023, 37, 0354-5180, 4683, 10.2298/FIL2314683M |
{A} | The previous one in Definition 2.17 [12] | The current method in Definition 2.17 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\delta\beta}}_R}(A) | |
\{a\} | \phi | \{a\} | \{a\} | 0 | \{a\} | \{a\} | \phi | 1 |
\{b\} | \phi | \{b\} | \{b\} | 0 | \{b\} | \{b\} | \phi | 1 |
\{c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{c\} | \{c\} | \phi | 1 |
\{a, b\} | \phi | \{a, b\} | \{a, b\} | 0 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{a, c\} | \{a, c\} | \phi | 1 |
\{b, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{b, c\} | \{b, c\} | \phi | 1 |
X | X | X | \phi | 1 | X | X | \phi | 1 |
A | Hosny's method 17 [12] | The current method in Definition 6.1 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | |
\{a\} | \phi | \{a\} | \phi | 0 | \phi | \{a\} | \{a\} | 0 |
\{b\} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} |
\{c\} | \{c\} | \{c\} | \phi | 1 | \{c\} | \{c\} | \phi | 1 |
\{d\} | \{d\} | \{d\} | \phi | 1 | \{d\} | \{d\} | \phi | 1 |
\{e\} | \phi | \{e\} | \{e\} | 0 | \{e\} | \{e\} | \phi | 1 |
\{a, b\} | \{a, b\} | \{a, b\} | \phi | 1 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} |
\{a, d\} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} |
\{a, e\} | \phi | \{a, e\} | \{a, e\} | 0 | \{e\} | \{a, e\} | \{a\} | \frac{1}{2} |
\{b, c\} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} |
\{b, d\} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} |
\{b, e\} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} |
\{c, d\} | \{c, d\} | \{c, d\} | \phi | 1 | \{c, d\} | \{c, d\} | \phi | 1 |
\{c, e\} | \{c, e\} | \{c, e\} | \phi | 1 | \{c, e\} | \{c, e\} | \phi | 1 |
\{d, e\} | \{d, e\} | \{d, e\} | \phi | 1 | \{d, e\} | \{d, e\} | \phi | 1 |
\{a, b, c\} | \{a, b, c\} | \{a, b, c\} | \phi | 1 | \{a, b, c\} | \{a, b, c\} | \phi | 1 |
\{a, b, d\} | \{a, b, d\} | \{a, b, d\} | \phi | 1 | \{a, b, d\} | \{a, b, d\} | \phi | 1 |
\{a, b, e\} | \{a, b, e\} | \{a, b, e\} | \phi | 1 | \{a, b, e\} | \{a, b, e\} | \phi | 1 |
\{a, c, d\} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} |
\{a, c, e\} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} |
\{a, d, e\} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} |
\{b, c, d\} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} |
\{b, c, e\} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} |
\{b, d, e\} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} |
\{c, d, e\} | \{c, d, e\} | \{c, d, e\} | \phi | 1 | \{c, d, e\} | \{c, d, e\} | \phi | 1 |
\{a, b, c, d\} | \{a, b, c, d\} | X | \{e\} | \frac{4}{5} | \{a, b, c, d\} | \{a, b, c, d\} | \phi | 1 |
\{a, b, c, e\} | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 |
\{a, b, d, e\} | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 |
\{a, c, d, e\} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} |
\{b, c, d, e\} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} |
X | X | X | \phi | 1 | X | X | \phi | 1 |
a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | |
\{x_{1}\} | 0.23 | 254.2 | 2.216 | -0.02 | 82.2 |
\{x_{2}\} | -0.48 | 303.6 | 2.994 | -1.24 | 112.3 |
\{x_{3}\} | -0.61 | 287.9 | 2.994 | -1.08 | 103.7 |
\{x_{4}\} | 0.45 | 282.9 | 2.933 | -0.11 | 99.1 |
\{x_{5}\} | -0.11 | 335.0 | 3.458 | -0.19 | 127.5 |
$$ | x_i\mathcal{R}_{1} | x_i\mathcal{R}_{2} | x_i\mathcal{R}_{3} | x_i\mathcal{R}_{4} | x_i\mathcal{R}_{5} |
\{x_{1}\} | \{x_1, x_4\} | X | X | \{x_1, x_4, x_5\} | X |
\{x_{2}\} | X | \{x_2, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_5\} |
\{x_{3}\} | X | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_3, x_4, x_5\} |
\{x_{4}\} | \{x_4\} | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | \{x_1, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} |
\{x_{5}\} | \{x_1, x_4, x_5\} | \{x_5\} | \{x_5\} | \{x_1, x_4, x_5\} | \{x_3, x_5\} |
Student | Science | German | Mathematics | Decision |
\{x_{1}\} | Bad | Good | Medium | Accept |
\{x_{2}\} | Good | Bad | Medium | Accept |
\{x_{3}\} | Good | Good | Good | Accept |
\{x_{4}\} | Bad | Good | Bad | Reject |
\{x_{5}\} | Good | Bad | Medium | Reject |
\{x_{6}\} | Bad | Good | Good | Accept |
{A} | The previous one in Definition 2.17 [12] | The current method in Definition 2.17 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\delta\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\delta\beta}}_R}(A) | |
\{a\} | \phi | \{a\} | \{a\} | 0 | \{a\} | \{a\} | \phi | 1 |
\{b\} | \phi | \{b\} | \{b\} | 0 | \{b\} | \{b\} | \phi | 1 |
\{c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{c\} | \{c\} | \phi | 1 |
\{a, b\} | \phi | \{a, b\} | \{a, b\} | 0 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{a, c\} | \{a, c\} | \phi | 1 |
\{b, c\} | \{c\} | X | \{a, b\} | \frac{1}{3} | \{b, c\} | \{b, c\} | \phi | 1 |
X | X | X | \phi | 1 | X | X | \phi | 1 |
A | Hosny's method 17 [12] | The current method in Definition 6.1 | ||||||
{\underline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\beta}}_R}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\beta}}_R}(A) | {{ACC}^{\mathcal{I}-{\beta}}_R}(A) | {\underline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {\overline{\mathcal{R}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{\mathcal{BND}}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | {{ACC}^{\mathcal{I}-{\bigwedge_{\beta}}_{R}}}(A) | |
\{a\} | \phi | \{a\} | \phi | 0 | \phi | \{a\} | \{a\} | 0 |
\{b\} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} | \{b\} | \{a, b\} | \{a\} | \frac{1}{2} |
\{c\} | \{c\} | \{c\} | \phi | 1 | \{c\} | \{c\} | \phi | 1 |
\{d\} | \{d\} | \{d\} | \phi | 1 | \{d\} | \{d\} | \phi | 1 |
\{e\} | \phi | \{e\} | \{e\} | 0 | \{e\} | \{e\} | \phi | 1 |
\{a, b\} | \{a, b\} | \{a, b\} | \phi | 1 | \{a, b\} | \{a, b\} | \phi | 1 |
\{a, c\} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} | \{c\} | \{a, c\} | \{a\} | \frac{1}{2} |
\{a, d\} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} | \{d\} | \{a, d\} | \{a\} | \frac{1}{2} |
\{a, e\} | \phi | \{a, e\} | \{a, e\} | 0 | \{e\} | \{a, e\} | \{a\} | \frac{1}{2} |
\{b, c\} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} | \{b, c\} | \{a, b, c\} | \{a\} | \frac{2}{3} |
\{b, d\} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} | \{b, d\} | \{a, b, d\} | \{a\} | \frac{2}{3} |
\{b, e\} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} | \{b, e\} | \{a, b, e\} | \{a\} | \frac{2}{3} |
\{c, d\} | \{c, d\} | \{c, d\} | \phi | 1 | \{c, d\} | \{c, d\} | \phi | 1 |
\{c, e\} | \{c, e\} | \{c, e\} | \phi | 1 | \{c, e\} | \{c, e\} | \phi | 1 |
\{d, e\} | \{d, e\} | \{d, e\} | \phi | 1 | \{d, e\} | \{d, e\} | \phi | 1 |
\{a, b, c\} | \{a, b, c\} | \{a, b, c\} | \phi | 1 | \{a, b, c\} | \{a, b, c\} | \phi | 1 |
\{a, b, d\} | \{a, b, d\} | \{a, b, d\} | \phi | 1 | \{a, b, d\} | \{a, b, d\} | \phi | 1 |
\{a, b, e\} | \{a, b, e\} | \{a, b, e\} | \phi | 1 | \{a, b, e\} | \{a, b, e\} | \phi | 1 |
\{a, c, d\} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} | \{c, d\} | \{a, c, d\} | \{a\} | \frac{2}{3} |
\{a, c, e\} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} | \{c, e\} | \{a, c, e\} | \{a\} | \frac{2}{3} |
\{a, d, e\} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} | \{d, e\} | \{a, d, e\} | \{a\} | \frac{2}{3} |
\{b, c, d\} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} | \{b, c, d\} | X | \{a, e\} | \frac{3}{5} |
\{b, c, e\} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} | \{b, c, e\} | \{a, b, c, e\} | \{a\} | \frac{3}{4} |
\{b, d, e\} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} | \{b, d, e\} | \{a, b, d, e\} | \{a\} | \frac{3}{4} |
\{c, d, e\} | \{c, d, e\} | \{c, d, e\} | \phi | 1 | \{c, d, e\} | \{c, d, e\} | \phi | 1 |
\{a, b, c, d\} | \{a, b, c, d\} | X | \{e\} | \frac{4}{5} | \{a, b, c, d\} | \{a, b, c, d\} | \phi | 1 |
\{a, b, c, e\} | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 | \{a, b, c, e\} | \{a, b, c, e\} | \phi | 1 |
\{a, b, d, e\} | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 | \{a, b, d, e\} | \{a, b, d, e\} | \phi | 1 |
\{a, c, d, e\} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} | \{c, d, e\} | \{a, c, d, e\} | \{a\} | \frac{3}{4} |
\{b, c, d, e\} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} | \{b, c, d, e\} | X | \{a\} | \frac{4}{5} |
X | X | X | \phi | 1 | X | X | \phi | 1 |
a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | |
\{x_{1}\} | 0.23 | 254.2 | 2.216 | -0.02 | 82.2 |
\{x_{2}\} | -0.48 | 303.6 | 2.994 | -1.24 | 112.3 |
\{x_{3}\} | -0.61 | 287.9 | 2.994 | -1.08 | 103.7 |
\{x_{4}\} | 0.45 | 282.9 | 2.933 | -0.11 | 99.1 |
\{x_{5}\} | -0.11 | 335.0 | 3.458 | -0.19 | 127.5 |
$$ | x_i\mathcal{R}_{1} | x_i\mathcal{R}_{2} | x_i\mathcal{R}_{3} | x_i\mathcal{R}_{4} | x_i\mathcal{R}_{5} |
\{x_{1}\} | \{x_1, x_4\} | X | X | \{x_1, x_4, x_5\} | X |
\{x_{2}\} | X | \{x_2, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_5\} |
\{x_{3}\} | X | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | X | \{x_2, x_3, x_4, x_5\} |
\{x_{4}\} | \{x_4\} | \{x_2, x_3, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} | \{x_1, x_4, x_5\} | \{x_2, x_3, x_4, x_5\} |
\{x_{5}\} | \{x_1, x_4, x_5\} | \{x_5\} | \{x_5\} | \{x_1, x_4, x_5\} | \{x_3, x_5\} |
Student | Science | German | Mathematics | Decision |
\{x_{1}\} | Bad | Good | Medium | Accept |
\{x_{2}\} | Good | Bad | Medium | Accept |
\{x_{3}\} | Good | Good | Good | Accept |
\{x_{4}\} | Bad | Good | Bad | Reject |
\{x_{5}\} | Good | Bad | Medium | Reject |
\{x_{6}\} | Bad | Good | Good | Accept |