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Design and experimentation of sampled-data controller in T-S fuzzy systems with input saturation through the use of linear switching methods

  • Received: 30 October 2023 Revised: 22 November 2023 Accepted: 26 November 2023 Published: 25 December 2023
  • MSC : 34A07, 34D20, 93D20

  • In this study, the stability and stabilization analyses are discussed for Takagi-Sugeno (T-S) fuzzy systems with input saturation. A fuzzy-based sampled-data control is designed to stabilize the T-S fuzzy systems. Based on the Lyapunov method and some integral inequality techniques, a set of sufficient conditions is obtained as linear matrix inequality (LMI) constraints to guarantee the asymptotic stability of the considered system. In this process, the linear switching method is utilized to design a controller that is dependent on the membership function, and an integral inequality is utilized. Additionally, determination of the controller parameters is achieved by resolving a series of LMI constraints. The effectiveness of these criteria is demonstrated through a real system that is modeled by the T-S system.

    Citation: YeongJae Kim, YongGwon Lee, SeungHoon Lee, Palanisamy Selvaraj, Ramalingam Sakthivel, OhMin Kwon. Design and experimentation of sampled-data controller in T-S fuzzy systems with input saturation through the use of linear switching methods[J]. AIMS Mathematics, 2024, 9(1): 2389-2410. doi: 10.3934/math.2024118

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  • In this study, the stability and stabilization analyses are discussed for Takagi-Sugeno (T-S) fuzzy systems with input saturation. A fuzzy-based sampled-data control is designed to stabilize the T-S fuzzy systems. Based on the Lyapunov method and some integral inequality techniques, a set of sufficient conditions is obtained as linear matrix inequality (LMI) constraints to guarantee the asymptotic stability of the considered system. In this process, the linear switching method is utilized to design a controller that is dependent on the membership function, and an integral inequality is utilized. Additionally, determination of the controller parameters is achieved by resolving a series of LMI constraints. The effectiveness of these criteria is demonstrated through a real system that is modeled by the T-S system.



    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. Iϕ, IP(X) is an ideal on X, if

    (i) AI and BIABI.

    (ii) AI and BABI.

    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 xX (J-nd) (nJ(x)),J{R,L,<R>,<L>,I,U,<I>,<U>} defined as:

    (i) R-nd: nR(x)={yX:xRy}.

    (ii) L-nd: nL(x)={yX:yRx}.

    (iii) <R>-nd: n<R>(x)=xnR(y)nR(y).

    (iv) <L>-nd: n<L>(x)=xnL(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:XP(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 AX. Then, τJ={AX:aA,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={FX:FτJ},F is the complement of F.

    Definition 2.4. [2] Let (X,R,ΞJ) be a J-ndS and AX. 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. AX 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. AX 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 AintJ[clJ(intJ(A))].

    (iv) βJ-open (semi preopen), if AclJ[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.

    Figure 1.  The relationships between τJ,ΓJ,ηJO(X) and ηJC(X).

    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 AX. 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 AX. The δ-J-closure of A is defined by clδJ(A)={xX:AintJ(clJ(G))ϕ,GτJ and xG}. 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)=XclδJ(XA).

    Definition 2.9. [11] Let (X,R,ΞJ) be a J-ndS and AX. A subset A is called δβJ-open, if AclJ[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 AX Then, the δβJ-lower, δβJ-upper approximations, δβJ-boundary and δβJ-accuracy of A are defined respectively by:

    R_δβJ(A)={GδβJO(X):GA}=δβJ-interior of A.

    ¯RδβJ(A)={HδβJC(X):AH}=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 AX. 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 AX. 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 AX. 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 AX. A subset βJ is defined as follows: βJ(A)={G:AG,GβJO(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 AX. A subset A is called βJ-set if A=βJ(A). The family of all βJ-set and βJ-set are denoted by τβJ and ΓβJ respectively.

    Proposition 2.2. [11] Every βJ-open set is βJ-set.

    Definition 2.14. [11] Let (X,R,ΞJ) be a J-ndS, AX. The βJ-lower, βJ-upper approximations, βJ-boundary and βJ-accuracy of A are defined respectively by:

    R_βJ(A)={GτβJ:GA}=βJ-interior of A.

    ¯RβJ(A)={HΓβJ:AH}=βJ-closure of A.

    BNDβJ(A)=¯RβJ(A)R_βJ(A).

    ACCβJ(A)=|R_βJ(A)||¯RβJ(A)|, where |¯RβJ(A)|0.

    Theorem 2.3. [11] Let (X,R,ΞJ) be a J-ndS and AX. 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) ¯RβJ(A)¯RβJ(A)¯RγJ(A)¯RpJ(A)¯RαJ(A).

    (v) ¯RβJ(A)¯RβJ(A)¯RγJ(A)¯RsJ(A)¯RαJ(A).

    (vi) ¯RβJ(A)¯RJ(A).

    Corollary 2.2. [11] Let (X,R,ΞJ) be a J-ndS and AX. Then

    (i) BNDβJ(A)BNDβJ(A)BNDγJ(A)BNDpJ(A)BNDαJ(A).

    (ii) BNDβJ(A)BNDβJ(A)BNDγJ(A)BNDsJ(A)BNDαJ(A).

    (iii) BNDβJ(A)BNDJ(A).

    (iv) ACCαJ(A)ACCpJ(A)ACCγJ(A)ACCβJ(A)ACCβJ(A).

    (v) ACCαJ(A)ACCsJ(A)ACCγJ(A)ACCβJ(A)ACCβJ(A).

    (vi) ACCJ(A)ACCβJ(A).

    Definition 2.15. [11] Let (X,R,ΞJ) be a J-ndS and AX. A subset A is called

    (i) βJ-definable (βJ-exact) if ¯RβJ(A)=R_βJ(A) or BNDβJ(A)=ϕ.

    (ii) βJ-rough if ¯RβJ(A)R_βJ(A) or BNDβJ(A)ϕ.

    Definition 2.16. [12] Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. AX is called

    (i) I-αJ-open, if GτJ such that (AintJ(clJ((G))I and (GA)I.

    (ii) I-J-Preopen (briefly I-PJ-open), if GτJ such that (AG)I and (GclJ(A))I.

    (iii) I-J-Semi open (briefly I-SJ-open), if GτJ such that (AclJ(G))I and (GA)I.

    (iv) I-βJ-open, if GτJ such that (AclJ(G))I and (GclJ(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:

    IαJopenIPJopenISJopenIβJopen.

    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)IαJO(IαJC)IPJO(IPJC)ISJO(ISJC)IβJO(IβJC).

    Definition 2.17. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. 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_IηJ(A)={GI-ηJO(X):GA}=I-J-nearly interior of A.

    ¯RIηJ(A)={HI-ηJC(X):AH}=I-J-nearly closure of A.

    BNDIηJ(A)=¯RIηJ(A)R_IηJ(A).

    ACCIηJ(A)=|R_IηJ(A)||¯RIηJ(A)|, where |¯RIηJ(A)|0.

    Definition 2.18. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. A is an I-ηJ-nearly definable (I-ηJ-nearly exact) set if ¯RIηJ(A)=R_IηJ(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 AX. Then

    (i) R_ηJ(A)R_IηJ(A).

    (ii) R_J(A)R_IηJ(A).

    (iii) ¯RIηJ(A)¯RηJ(A).

    (iv) ¯RIηJ(A)¯RJ(A).

    Corollary 2.3. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) BNDIηJ(A)BNDηJ(A).

    (ii) BNDηJ(A)BNDJ(A).

    (iii) ACCηJ(A)ACCIηJ(A).

    (iv) ACCJ(A)ACCIηJ(A).

    Proposition 2.5. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) R_IPJ(A)R_IβJ(A).

    (ii) R_IαJ(A)R_ISJ(A)R_IβJ(A).

    (iii) ¯RIβJ(A)¯RIPJ(A).

    (iv) ¯RIβJ(A)¯RISJ(A)¯RIαJ(A).

    Corollary 2.4. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) BNDIβJ(A)BNDIPJ(A).

    (ii) BNDIβJ(A)BNDISJ(A)BNDIαJ(A).

    (iii) ACCIPJ(A)ACCIβJ(A).

    (iv) ACCIαJ(A)ACCISJ(A)ACCIβJ(A).

    Definition 2.19. [28] Let R be an equivalence relation on X and AX. Then the rough membership functions of AX are defined as μA:X[0,1], where

    μA(x)=|[x]RA||[x]R|,xX.

    [x]R denotes to an equivalence classes.

    Definition 2.20. [3] Let (X,R,ΞJ) be a J-ndS, AX and xX. 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, AX and xX. Then the J-rough nearly membership functions of A are defined by μηJA[0,1], where

    μηJA(x)={1if1ψηJA(x).min(ψηJA(x))otherwise.}.andψηJA(x)=|ηJ(x)A||ηJ(x)|,xηJ(x),ηJ(x)ηJO(X).

    Definition 2.22. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X,AX and xX. The IJ-nearly rough membership functions of A are defined by μIηJA[0,1], where

    μIηJA(x)={1if1ψIηJA(x).min(ψIηJA(x))otherwise.}.andψIηJA(x)=|IηJ(x)A||IηJ(x)|,xIηJ(x),IηJ(x)IηJO(X).

    Lemma 2.1. [12] Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) μJA(x)=1μηJA(x)=1μIηJA(x)=1,xX.

    (ii) μJA(x)=0μηJA(x)=0μIηJA(x)=0,xX.

    Definition 2.23. [3] Let (X,R,ΞJ) be a J-ndS, xX and AX:

    (i) If xR_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 xR_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,xX and AX:

    (i) If xR_IηJ(A), then x is J-nearly surely with respect to I (IηJ-surely) belongs to A, denoted by x_IηJA.

    (ii) If x¯RIηJ(A), then x is J-nearly possibly with respect to I (briefly IηJ-possibly) belongs to A, denoted by x¯IηJA.

    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 AX. Then

    (i) If x_JAx_ηJAx_IηJA.

    (ii) If x¯IηJAx¯ηJAx¯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. AX is called I-δβJ-open, if GτJ such that (AclJ(G))I and (GclδJ(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:

    IαJopen         IPJopen                                  ISJopen         IβJopen         IδβJopen.

    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)IαJO(IαJC)    IPJO(IPJC)                                                                     ISJO(ISJC)IβJO(IβJC)IδβJO(IδβJC).

    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 (AclJ(G))I,(GclδJ(A))I,(BclJ(H))I and (HclδJ(B))I. Hence, (GclδJ(AB))(GclδJ(A))I, (HclδJ(AB))(HclδJ(B))I and so, (GclδJ(AB))(HclδJ(AB))I. Let W=GH, then (WclδJ(AB))I. Also, (AclJ(W))(AclJ(G))I and (BclJ(W))(BclJ(H))I. Then, (AclJ(W))(BclJ(W))(AclJ(G))(BclJ(H))I and so ((AB)clJ(W))(AclJ(G))(BclJ(H))I. Thus, AB 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 AB={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 AX. 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)I-δβRO(X).

    I-δβIO(X)I-δβRO(X).

    I-δβUO(X)I-δβLO(X).

    I-δβIO(X)I-δβLO(X).

    I-δβ<U>O(X)I-δβ<L>O(X).

    I-δβ<I>O(X)I-δβ<L>O(X).

    I-δβRO(X) is not the dual of I-δβLO(X) and I-δβ<R>O(X) is not the dual of I-δβ<L>O(X).

    ● In a similar way, I can add examples to show that I-δβLO(X)I-δβIO(X),I-δβRO(X)I-δβUO(X),I-δβRO(X)I-δβIO(X),I-δβLO(X)I-δβUO(X),I-δβIO(X)I-δβLO(X),I-δβ<L>O(X)I-δβ<I>O(X),I-δβ<L>O(X)I-δβ<U>O(X),I-δβ<R>O(X)I-δβ<I>O(X) and I-δβ<U>O(X)I-δβ<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 I-δβJ-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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. The I-δβJ-lower, I-δβJ-upper approximations, I-δβJ-boundary regions and I-δβJ-accuracy of A are defined respectively by:

    R_IδβJ(A)={GI-δβJO(X):GA}=I-δβJ-interior of A.

    ¯RIδβJ(A)={HI-δβJC(X):AH}=I-δβJ-closure of A.

    BNDIδβJ(A)=¯RIδβJ(A)R_IδβJ(A).

    ACCIδβJ(A)=|R_IδβJ(A)||¯RIδβJ(A)|, where |¯RIδβJ(A)|0.

    The following proposition presents the main properties of the current I-δβJ-lower and I-δβJ-upper approximations.

    Proposition 4.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A,BX. Then,

    (i) R_IδβJ(A)A¯RIδβJ(A) equality hold if A=ϕ or X.

    (ii) AB¯RIδβJ(A)¯RIδβJ(B).

    (iii) ABR_IδβJ(A)R_IδβJ(B).

    (iv) ¯RIδβJ(AB)¯RIδβJ(A)¯RIδβJ(B).

    (v) R_IδβJ(AB)R_IδβJ(A)R_IδβJ(B).

    (vi) ¯RIδβJ(AB)¯RIδβJ(A)¯RIδβJ(B).

    (vii) R_IδβJ(AB)R_IδβJ(A)R_IδβJ(B).

    (viii) R_IδβJ(A)=(¯RIδβJ(A)), ¯RIδβJ(A)=(R_IδβJ(A)).

    (ix) ¯RIδβJ(¯RIδβJ(A))=¯RIδβJ(A).

    (x) R_IδβJ(R_IδβJ(A))=R_IδβJ(A).

    (xi) R_IδβJ(R_IδβJ(A))¯RIδβJ(R_IδβJ(A)).

    (xii) R_IδβJ(¯RIδβJ(A))¯RIδβJ(¯RIδβJ(A)).

    (xiii) x¯RIδβJ(A)GAϕ,GI-δβJO(X),xG.

    (xiv) xR_IδβJ(A)GI-δβJO(X),xG,GA.

    The proof of this proposition is simple using the I-δβJ-interior and I-δβJ-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},¯RIδβR(A)=X, then ¯RIδβR(A)A, take A={e},R_IδβR(A)=ϕ. Then, AR_IδβR(A).

    (ii) For part (iv), if A={a,b,c,d},B={b,c,d,e},AB={b,c,d},¯RIδβR(A)=X,¯RIδβR(B)=B,¯RδβR(AB)=AB, then ¯RIδβR(A)¯RIδβR(B)={b,c,d,e}{b,c,d}=¯RIδβR(AB).

    (iii) For part (v), if A={a},B={e},AB={a,e},R_IδβR(A)=A,R_IδβR(B)=ϕ,R_IδβR(AB)=AB, then R_IδβR(AB)={a,e}{a}=R_IδβR(A)R_IδβR(B).

    (iv) For part (vi), if A={a,c},B={b,d},AB={a,b,c,d},¯RIδβR(A)=A,¯RIδβR(B)=B,¯RIδβR(AB)=X, then ¯RIδβR(AB)=X{a,b,c,d}=¯RIδβR(A)¯RIδβR(B).

    (v)For part (vii), if A={a,e},B={c,e},AB={e},R_IδβR(A)=A,R_IδβR(B)=B,R_IδβR(AB)=ϕ, then R_IδβR(A)R_IδβR(B)={e}ϕ=R_IδβR(AB).

    (vi) For part (xi), if A={a,b,c,d},R_IδβR(R_IδβR(A))=A,¯RIδβR(R_IδβR(A))=X, then ¯RIδβR(R_IδβR(A))R_IδβR(R_IδβR(A)).

    (vii) For part (xii), if A={e},¯RIδβR(¯RIδβR(A))=A,R_IδβR(¯RIδβR(A))=ϕ, then ¯RIδβR(¯RIδβR(A))R_IδβR(¯RIδβR(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 ¯RIδβR(A)=A,¯RIδβR(B)=X. Therefore, ¯RIδβR(A)¯RIδβR(B), but AB.

    (ii) For part (iii), if A={e},B={c,d}, then R_IδβR(A)=ϕ,R_IδβR(B)=B. Therefore, R_IδβR(A)R_IδβR(B), but AB.

    Definition 4.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. A is an I-δβJ-definable (an I-δβJ-exact) set if ¯RIδβJ(A)=R_IδβJ(A). Otherwise, A is an I-δβJ-rough set.

    In Example 3.3 A={c} is I-δβR-exact, while B={e} is I-δβR-rough.

    Remark 4.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then the intersection of two I-δβR-exact sets does not need to be an I-δβR-exact set as in Example 3.3 {a,e} and {c,e} are I-δβR-exact sets, but {a,e}{c,e}={e} is not an I-δβR-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,R,ΞJ) be a J-ndS and AX. Then

    (i) R_αJ(A)R_pJ(A)R_γJ(A)R_βJ(A)R_δβJ(A)R_IδβJ(A).

    (ii) R_αJ(A)R_sJ(A)R_γJ(A)R_βJ(A)R_δβJ(A)R_IδβJ(A).

    (iii) R_J(A)R_IδβJ(A).

    (iv) ¯RIδβJ(A)¯RδβJ(A)¯RβJ(A)¯RγJ(A)¯RpJ(A)¯RαJ(A).

    (v) ¯RIδβJ(A)¯RδβJ(A)¯RβJ(A)¯RγJ(A)¯RsJ(A)¯RαJ(A).

    (vi) ¯RIδβJ(A)¯RJ(A).

    Proof. (i) By Theorem 2.2 [11], R_αJ(A)R_pJ(A)R_γJ(A)R_βJ(A)R_δβJ(A), and R_δβJ(A)={GδβJO(X):GA}{GI-δβJO(X):GA}=R_IδβJ(A) (by Proposition 3.1).

    (ii) It is similar to (i).

    (iii) By Theorem 2.2 [11], R_J(A)R_δβJ(A), and by (1) R_δβJ(A)R_IδβJ(A). Hence, R_J(A)R_IδβJ(A).

    (iv)–(vi) They are similar to (i)–(iii).

    Corollary 4.1. Let (X,R,ΞJ) be a J-ndS and AX. Then

    (i) BNDIδβJ(A)BNDδβJ(A)BNDβJ(A)BNDγJ(A)BNDpJ(A)BNDαJ(A).

    (ii) BNDIδβJ(A)BNDδβJ(A)BNDβJ(A)BNDγJ(A)BNDsJ(A)BNDαJ(A).

    (iii) BNDIδβJ(A)BNDJ(A).

    (iv) ACCαJ(A)ACCpJ(A)ACCγJ(A)ACCβJ(A)ACCδβJ(A)ACCIδβJ(A).

    (v) ACCαJ(A)ACCsJ(A)ACCγJ(A)ACCβJ(A)ACCδβJ(A)ACCIδβJ(A).

    (vi) ACCJ(A)ACCIδβJ(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 R_R(A)=R_δβR(A)=ϕ,R_IδβR(A)={d} and if A={a,b,c}, then ¯RR(A)=¯RδβR(A)=X,¯RIδβR(A)={a,b,c}. Moreover, take A={a,b,c}, then the boundary and accuracy by the present method in Definition 4.1 are ϕ 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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) A is αJ-exact A is SJ-exact A is βJ-exact δβJ-exact A is I-δβJ-exact.

    (ii) A is PJ-exact A is βJ-exact A is δβJ-exact A is I-δβJ-exact.

    (iii) A is J-exact A is I-δβJ-exact.

    (iv) A is I-δβJ-rough A is δβJ-rough A is βJ-rough A is SJ-rough A is αJ-rough.

    (v) A is I-δβJ-rough A is δβJ-rough A is βJ-rough A is PJ-rough.

    (vi) A is I-δβJ-rough 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 I-δβR-exact, but it is neither δβR-exact nor R-exact.

    (ii) If A={a,b,c}, then it is R-rough and δβR-rough, but it is not I-δβR-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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) R_IPJ(A)R_IβJ(A)R_IδβJ(A).

    (ii) R_IαJ(A)R_ISJ(A)R_IβJ(A)R_IδβJ(A).

    (iii) ¯RIδβJ(A)¯RIβJ(A)¯RIPJ(A).

    (iv) ¯RIδβJ(A)¯RIβJ(A)¯RISJ(A)¯RIαJ(A).

    Proof. By Proposition 3.2, the proof is obvious.

    Corollary 4.3. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) BNDIδβJ(A)BNDIβJ(A)BNDIPJ(A).

    (ii) BNDIδβJ(A)BNDIβJ(A)BNDISJ(A)BNDIαJ(A).

    (iii) ACCIPJ(A)ACCIβJ(A)ACCIδβJ(A).

    (iv) ACCIαJ(A)ACCISJ(A)ACCIβJ(A)ACCIδβJ(A).

    Corollary 4.4. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) A is I-αJ-exact A is I-SJ-exact A is I-βJ-exact A is I-δβJ-exact.

    (ii) A is I-PJ-exact A is I-βJ-exact A is I-δβJ-exact.

    (iii) A is I-δβJ-rough A is I-βJ-rough A is I-SJ-rough A is I-αJ-rough.

    (iv) A is I-δβJ-rough A is I-βJ-rough A is I-PJ-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.

    Table 1.  Comparison between the boundary and accuracy by using the current approximations in Definition 4.1 and the previous one in Definition 2.17 [12].
    {A} The previous one in Definition 2.17 [12] The current method in Definition 2.17
    R_IβR(A) ¯RIβR(A) BNDIβR(A) ACCIβR(A) R_IδβR(A) ¯RIδβR(A) BNDIδβR(A) ACCIδβR(A)
    {a} ϕ {a} {a} 0 {a} {a} ϕ 1
    {b} ϕ {b} {b} 0 {b} {b} ϕ 1
    {c} {c} X {a,b} 13 {c} {c} ϕ 1
    {a,b} ϕ {a,b} {a,b} 0 {a,b} {a,b} ϕ 1
    {a,c} {c} X {a,b} 13 {a,c} {a,c} ϕ 1
    {b,c} {c} X {a,b} 13 {b,c} {b,c} ϕ 1
    X X X ϕ 1 X X ϕ 1

     | Show Table
    DownLoad: CSV

    For example, take A={a,b}, then the boundary and accuracy by the present method in Definition 4.1 are ϕ 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 I-δβR-exact, but it is not I-βR-exact and consequently, not I-SR-exact, not I-αR-exact and not I-PR-exact, also A={a}, is I-βR-rough, but not I-δβR-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 I-δβJ-lower approximations and decreasing the I-δβJ-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 I-J-nearly open sets is developed and extended in this section by proposing the concept of I-βJ-sets. The main characterizations of this concept and the connections among them are investigated and analyzed. The concepts of I-βJ-sets and I-δβJ-open sets are different and independent (see Remark 5.5).

    Definition 5.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. A subset I-βJ is defined as follows: I-βJ(A)={G:AG,GI-βJO(X)}. The complement of I-βJ(A)-set is called I-βJ(A)-set.

    In the following lemma I summarize the fundamental properties of the subset I-βJ.

    Lemma 5.1. For subsets A,B and Aα(αΔ) of a J-ndS (X,R,ΞJ), the following implications hold:

    (i) AI-βJ(A).

    (ii) If AB, then I-βJ(A)I-βJ(B).

    (iii) I-βJ(I-βJ(A))=I-βJ(A).

    (iv) If AI-βJO(X), then A=I-βJ(A). (v) I-βJ({Aα:αΔ})={I-βJ(Aα):αΔ}.

    (vi) I-βJ({Aα:αΔ}){I-βJ(Aα):αΔ}.

    Proof. I prove only (v) and (vi) since the other are consequences of Definition 5.1.

    (v) First for each αΔ,I-βJ(Aα)I-βJ(αΔAα). Hence, αΔI-βJ(Aα)I-βJ(αΔAα). Conversely, suppose that xαΔI-βJ(Aα). Then, xI-βJ(Aα) for each αΔ and hence there exists GαI-βJO(X) such that AαGα and xGα for each αΔ. I have that αΔAααΔGα and αΔGα is I-βJ-open set which does not contain x. Therefore, xI-βJ(αΔAα). Thus, I-βJ(αΔAα)αΔI-βJ(Aα).

    (vi) Suppose that, x{I-βJ(Aα):αΔ}. There exists α0Δ such that xI-βJ(Aα0) and there exists I-βJ-open set G such that xG and Aα0G. I have that αΔAαAα0G and xG. Therefore, xI-βJ({Aα:αΔ}).

    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 I-βJ(A)={a,b} and I-βJ(A)A.

    (ii) For part (vi), if A={b} and B={a}, then AB=ϕ and I-βJ(A)={b},I-βJ(B)={a,b},I-βJ(AB)=ϕ and I-βJ(A)I-βJ(B)={b}I-βJ(AB)=ϕ.

    (iii) For part (ii), if A={a} and B={b}, then I-βJ(A)={a,b} and I-βJ(B)={b}. Therefore, I-β(A)I-βJ(B), but AB.

    Definition 5.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. A subset A is called an I-βJ-set if A=IβJ(A). The family of all I-βJ-sets and I-βJ-sets are denoted by τIβJ and ΓIβJ respectively.

    Example 5.1. In Example 3.1, τIβR=P(X), in Example 3.2, τIβR=P(X){{a},{b},{a,b}} and in Example 3.2, τIβR={X,ϕ,{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 I-βJ-sets is an extension of the concept of βJ-sets.

    Proposition 5.1. Every βJ-set is I-βJ-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, PJ-open, SJ-open and αJ-open sets.

    (ii) The converse of Proposition 5.1 is not necessarily true as shown in the following example.

    Example 5.2. Let

    X={a,b,c,d},I={ϕ,{c}}

    and

    R={(a,a),(a,c),(b,a),(b,c),(c,c),(d,d)}

    be a binary relation defined on X thus aR=bR={a,c},cR={c} and dR={d}. Then, the topology associated with this relation is τR={X,ϕ,{c},{d},{a,c},{c,d},{a,c,d}}. It is clear that τIβR=P(X) and τβR={X,ϕ,{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,R,ΞJ) be a J-ndS and I be an ideal on X. If I={ϕ} in the current Definition 5.2, then I get Hosny 's Definition 2.13 [11].

    Proof. Straightforward.

    The following proposition shows that I-βJ-sets are generalization of I-βJ-open sets in Definition 2.16 [12]. Consequently, it is 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 5.2. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:

    IαJopen      IPJopen                          ISJopen            IβJopen         IβJset.

    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 I-βR-set, but it is not an I-βR-open set.

    Proposition 5.3. Let (X,R,ΞJ) be a J-ndS and I be an ideal on X. Then, the following implications hold:

    τJ(ΓJ)IαJO(IαJC)    IPJO(IPJC)                                                                            ISJO(ISJC)IβJO(IβJC)τIβJ(ΓIβJ).

    Proof. By Propositions 2.4 [12] and 5.2, the proof is obvious.

    In the following lemma I summarize the fundamental properties of I-βJ-sets.

    Lemma 5.2. For subsets A,B and Aα(αΔ) of a J-ndS (X,R,ΞJ), the following implications hold:

    (i) X,ϕ are I-βJ-sets.

    (ii) If Aα is an I-βJ-set αΔ, then αΔAα is an I-βJ-set.

    (iii) If Aα is an I-βJ-set αΔ, then αΔAα is an I-βJ-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 I-βJ-sets forms a topology.

    Remark 5.5. The I-δβJ-open sets of Definition 3.1 and the current Definition 5.2 of I-βJ-sets are different and independent. Example 3.3 shows that {a} is an I-δβJ-open set, but it is not an I-βJ-set. Moreover, it shows that {e} is an I-βJ-set, but it is not an I-δβJ-open set.

    Remark 5.6. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then the following statements are not true in general:

    (i) τIβUτIβRτIβI.

    (ii) τIβUτIβLτIβI.

    (iii) τIβ<U>τIβ<R>τIβ<I>.

    (iv) τIβ<U>τIβ<L>τIβ<I>.

    (v) τIβR is the dual of τIβL.

    (vi) τIβ<R> is the dual of τIβ<L>.

    So, the relationships among I-βJ-sets are not comparable as in Example 3.3:

    (i) τIβR={X,ϕ,{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) τIβL(X)={X,ϕ,{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) τIβI(X)=P(X).

    (iv) τIβU(X)=P(X).

    (v) τIβ<R>(X)={X,ϕ,{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) τIβ<L>(X)=P(X).

    (vii) τIβ<I>(X)=P(X).

    (viii) τIβ<U>(X)={X,ϕ,{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

    τIβU(X)τIβR(X).

    τIβI(X)τIβR(X).

    τIβU(X)τIβLX).

    τIβI(X)τIβLX).

    τIβ<U>(X)τIβ<R>(X).

    τIβ<R>(X)τIβ<U>(X).

    τIβ<L>(X)τIβ<U>(X).

    τIβ<I>(X)τIβ<U>(X).

    τIβ<L>(X)τIβ<I>(X).

    τIβ<I>(X)τIβ<L>(X).

    τIβR(X) is not the dual of τIβL and τIβ<R>(X) is not the dual of τIβ<L>(X).

    ● In a similar way, I can add examples to show that, τIβL(X)τIβI(X),τIβR(X)τIβI(X),τIβR(X)τIβU(X),τIβ<R>(X)τIβ<I>(X),τIβ<U>(X)τIβ<I>(X),τIβ<U>(X)τIβ<L>(X) and τIβ<L>(X)τIβ<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 I-βJ-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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. The I-βJ-lower, I-βJ-upper approximations, I-βJ-boundary regions and I-βJ-accuracy of A are defined respectively by:

    R_IβJ(A)={GτIβJ:GA}=I-βJ-interior of A.

    ¯RIβJ(A)={HΓIβJ:AH}=I-βJ-closure of A.

    BNDIβJ(A)=¯RIβJ(A)R_IβJ(A).

    ACCIβJ(A)=|R_IβJ(A)||¯RIβJ(A)|, where |¯RIβJ(A)|0.

    The following proposition studies the main properties of the current I-βJ-lower and I-βJ-upper approximations.

    Proposition 6.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A,BX. Then,

    (i) R_IβJ(A)</italic><italic>⊆A¯RIβJ(A) equality hold if A=ϕ or X.

    (ii) AB¯RIβJ(A)¯RIβJ(B).

    (iii) ABR_IβJ(A)R_IβJ(B).

    (iv) ¯RIβJ(AB)¯RIβJ(A)¯RIβJ(B).

    (v) R_IβJ(AB)R_IβJ(A)R_IβJ(B).

    (vi) ¯RIβJ(AB)=¯RIβJ(A)¯RIβJ(B).

    (vii) R_IβJ(AB)=R_IβJ(A)R_IβJ(B).

    (viii) R_IβJ(A)=(¯RIβJ(A)), ¯RIβJ(A)=(R_IβJ(A)).

    (ix) ¯RIβJ(¯RIβJ(A))=¯RIβJ(A).

    (x) R_IβJ(R_IβJ(A))=R_IβJ(A).

    (xi) R_IβJ(R_IβJ(A))¯RIβJ(R_IβJ(A)).

    (xii) R_IβJ(¯RIβJ(A))¯RIβJ(¯RIβJ(A)).

    (xiii) x¯RIβJ(A)GAϕ,GτIβJ,xG.

    (xiv) xR_IβJ(A)GτIβJ,xG,GA.

    The proof of this proposition is simple using I-βJ-interior and I-βJ-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},¯RIβR(A)={a,b,c,e}, then ¯RIβR(A)A, take A={a},R_IβR(A)=ϕ. Then, AR_IβR(A).

    (ii) For part (iv), if A={b,c,d,e},B={a,c,d,e},AB={c,d,e},¯RIβR(A)=X,¯RIβR(B)=B,¯RIβR(AB)=AB, then ¯RIβR(A)¯RIβR(B)={a,c,d,e}{c,d,e}=¯RIβR(AB).

    (iii) For part (v), if A={a},B={b},AB={a,b},R_IβR(A)=ϕ,R_IβR(B)=B,R_IβR(AB)=AB, then R_IβR(AB)={a,b}{b}=R_IβR(A)R_IβR(B).

    (iv) For part (xi), if A={b,c,d,e},R_IβR(R_IβR(A))=A,¯RIβR(R_IβR(A))=X, then ¯RIβR(R_IβR(A))R_IβR(R_IβR(A)).

    (v) For part (xii), if A={a},¯RIβR(¯RIβR(A))=A,R_IβR(¯RIβR(A))=ϕ, then ¯RIβR(¯RIβR(A))R_IβR(¯RIβR(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 ¯RIβR(A)=A,¯RIβR(B)=X. Therefore, ¯RIβR(A)¯RIβR(B), but AB.

    (ii) For part (iii), if A={a},B={c,d,e}, then R_IβR(A)=ϕ,R_IβR(B)=B. Therefore, R_IβR(A)R_IβR(B), but AB.

    Definition 6.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. A is an I-βJ-definable (I-βJ-exact) set if ¯RIβJ(A)=R_IβJ(A). Otherwise, A is an I-βJ-rough set.

    In Example 3.3 A={c} is I-βR-exact, while B={a} is I-βR-rough.

    Remark 6.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then the intersection of two I-βJ-rough sets does not need to be an I-βJ-rough set as in Example 3.3, {c,d} and {c,e}, are I-βR-rough sets, but {c,d}{c,e}={c} is not an I-βR-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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) R_αJ(A)R_pJ(A)R_γJ(A)R_βJ(A)R_βJ(A)R_IβJ(A).

    (ii) R_αJ(A)R_sJ(A)R_γJ(A)R_βJ(A)R_βJ(A)R_IβJ(A).

    (iii) R_J(A)R_IβJ(A).

    (iv) ¯RIβJ(A)¯RβJ(A)¯RβJ(A)¯RγJ(A)¯RpJ(A)¯RαJ(A).

    (v) ¯RIβJ(A)¯RβJ(A)¯RβJ(A)¯RγJ(A)¯RsJ(A)¯RαJ(A).

    (vi) ¯RIβJ(A)¯RJ(A).

    Proof. (i) By Theorem 2.3 [11], R_αJ(A)R_pJ(A)R_γJ(A)R_βJ(A)R_βJ(A), and R_βJ(A)={GτβJ:GA:GA}{GτIβJ:GA}=R_IβJ(A) (by Proposition 5.1).

    (ii) It is similar to (i).

    (iii) By Theorem 2.3 [11], R_J(A)R_βJ(A), and by (1) R_βJ(A)R_IβJ(A). Hence, R_J(A)R_IβJ(A).

    (iv)–(vi) They are similar to (i)–(iii).

    Corollary 6.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) BNDIβJ(A)BNDβJ(A)BNDβJ(A)BNDγJ(A)BNDpJ(A)BNDαJ(A).

    (ii) BNDIβJ(A)BNDβJ(A)BNDβJ(A)BNDγJ(A)BNDsJ(A)BNDαJ(A).

    (iii) BNDIβJ(A)BNDJ(A).

    (iv) ACCαJ(A)ACCpJ(A)ACCγJ(A)ACCβJ(A)ACCβJ(A)ACCIβJ(A).

    (v) ACCαJ(A)ACCsJ(A)ACCγJ(A)ACCβJ(A)ACCβJ(A)ACCIβJ(A).

    (vi) ACCJ(A)ACCIβJ(A).

    Corollary 6.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) A is αJ-exact A is SJ-exact A is βJ-exact A is βJ-exact A is I-βJ-exact.

    (ii) A is PJ-exact A is βJ-exact A is βJ-exact A is I-βJ-exact.

    (iii) A is J-exact A is I-βJ-exact.

    (iv) A is I-βJ-rough A is βJ-rough A is βJ-rough A is SJ-rough A is αJ-rough.

    (v) A is I-βJ-rough A is βJ-rough A is βJ-rough A is PJ-rough.

    (vi) A is I-βJ-rough 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 I-βR-exact, but it is neither βR-exact nor R-exact.

    (ii) If A={b}, then it is R-rough and βR-rough, but it is not I-βR-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,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) R_IPJ(A)R_IβJ(A)R_IβJ(A).

    (ii) R_IαJ(A)R_ISJ(A)R_IβJ(A)R_IβJ(A).

    (iii) ¯RIβJ(A)¯RIβJ(A)¯RIPJ(A).

    (iv) ¯RIβJ(A)¯RIβJ(A)¯RISJ(A)¯RIαJ(A).

    Proof. By Proposition 5.2, the proof is obvious.

    Corollary 6.3. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) BNDIβJ(A)BNDIβJ(A)BNDIPJ(A).

    (ii) BNDIβJ(A)BNDIβJ(A)BNDISJ(A)BNDIαJ(A).

    (iii) ACCIPJ(A)ACCIβJ(A)ACCIβJ(A).

    (iv) ACCIαJ(A)ACCISJ(A)ACCIβJ(A)ACCIβJ(A).

    Corollary 6.4. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) A is J-exact A is I-αJ-exact A isI-SJ-exact A is I-βJ-exact A is I-βJ-exact.

    (ii) A is I-PJ-exact A is I-βJ-exact A is I-βJ-exact.

    (iii) A is I-βJ-rough A is I-βJ-rough A is I-SJ-rough A is I-αJ-rough.

    (iv) A is I-βJ-rough A is I-βJ-rough A is I-PJ-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.

    Table 2.  Comparison between the boundary and accuracy by Hosny's method 2.17 [12] and the current approximations in Definition 6.1.
    A Hosny's method 17 [12] The current method in Definition 6.1
    R_IβR(A) ¯RIβR(A) BNDIβR(A) ACCIβR(A) R_IβR(A) ¯RIβR(A) BNDIβR(A) ACCIβR(A)
    {a} ϕ {a} ϕ 0 ϕ {a} {a} 0
    {b} {b} {a,b} {a} 12 {b} {a,b} {a} 12
    {c} {c} {c} ϕ 1 {c} {c} ϕ 1
    {d} {d} {d} ϕ 1 {d} {d} ϕ 1
    {e} ϕ {e} {e} 0 {e} {e} ϕ 1
    {a,b} {a,b} {a,b} ϕ 1 {a,b} {a,b} ϕ 1
    {a,c} {c} {a,c} {a} 12 {c} {a,c} {a} 12
    {a,d} {d} {a,d} {a} 12 {d} {a,d} {a} 12
    {a,e} ϕ {a,e} {a,e} 0 {e} {a,e} {a} 12
    {b,c} {b,c} {a,b,c} {a} 23 {b,c} {a,b,c} {a} 23
    {b,d} {b,d} {a,b,d} {a} 23 {b,d} {a,b,d} {a} 23
    {b,e} {b,e} {a,b,e} {a} 23 {b,e} {a,b,e} {a} 23
    {c,d} {c,d} {c,d} ϕ 1 {c,d} {c,d} ϕ 1
    {c,e} {c,e} {c,e} ϕ 1 {c,e} {c,e} ϕ 1
    {d,e} {d,e} {d,e} ϕ 1 {d,e} {d,e} ϕ 1
    {a,b,c} {a,b,c} {a,b,c} ϕ 1 {a,b,c} {a,b,c} ϕ 1
    {a,b,d} {a,b,d} {a,b,d} ϕ 1 {a,b,d} {a,b,d} ϕ 1
    {a,b,e} {a,b,e} {a,b,e} ϕ 1 {a,b,e} {a,b,e} ϕ 1
    {a,c,d} {c,d} {a,c,d} {a} 23 {c,d} {a,c,d} {a} 23
    {a,c,e} {c,e} {a,c,e} {a} 23 {c,e} {a,c,e} {a} 23
    {a,d,e} {d,e} {a,d,e} {a} 23 {d,e} {a,d,e} {a} 23
    {b,c,d} {b,c,d} X {a,e} 35 {b,c,d} X {a,e} 35
    {b,c,e} {b,c,e} {a,b,c,e} {a} 34 {b,c,e} {a,b,c,e} {a} 34
    {b,d,e} {b,d,e} {a,b,d,e} {a} 34 {b,d,e} {a,b,d,e} {a} 34
    {c,d,e} {c,d,e} {c,d,e} ϕ 1 {c,d,e} {c,d,e} ϕ 1
    {a,b,c,d} {a,b,c,d} X {e} 45 {a,b,c,d} {a,b,c,d} ϕ 1
    {a,b,c,e} {a,b,c,e} {a,b,c,e} ϕ 1 {a,b,c,e} {a,b,c,e} ϕ 1
    {a,b,d,e} {a,b,d,e} {a,b,d,e} ϕ 1 {a,b,d,e} {a,b,d,e} ϕ 1
    {a,c,d,e} {c,d,e} {a,c,d,e} {a} 34 {c,d,e} {a,c,d,e} {a} 34
    {b,c,d,e} {b,c,d,e} X {a} 45 {b,c,d,e} X {a} 45
    X X X ϕ 1 X X ϕ 1

     | Show Table
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    For example, take A={e}, then the boundary and accuracy by the present method in Definition 6.1 are ϕ 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 I-βJ-approximations in this section and the I-δβJ-approximations in Section 4 are different and independent. As, the concepts of I-δβJ-open sets and I-βJ-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 I-δβJ-rough membership functions and I-βJ-rough membership functions.

    Definition 7.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X,xX and AX:

    (i) If xR_IδβJ(A), then x is J-δβ-surely with respect to I (IδβJ-surely) belongs to A, denoted by x_IδβJA.

    (ii) If x¯RIδβJ(A), then x is J-δβ-possibly with respect to I (briefly IδβJ-possibly) belongs to A, denoted by x¯IδβJA.

    (iii) If xR_IβJ(A), then x is J-β-surely with respect to I (IβJ-surely) belongs to A, denoted by x_IδβJA.

    (iv) If x¯RIβJ(A), then x is J-β-possibly with respect to I (briefly IβJ-possibly) belongs to A, denoted by x¯IβJA.

    It is called J-δβ-strong (J-β-strong) and J-δβ-weak (J-β-weak) membership relations with respect to I respectively.

    Remark 7.1. According to Definitions 4.1 and 6.1, the I-δβJ-lower, I-δβJ-upper approximations, I-βJ-lower and I-βJ-upper approximations for any AX can be written as:

    (i) R_IδβJ(A)={xX:x_IδβJA}.

    (ii) ¯RIδβJ(A)={xX:x¯IδβJA}.

    (iii) R_IβJ(A)={xX:x_IβJA}.

    (iv) ¯RIβJ(A)={xX:x¯IβJA}.

    Lemma 7.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) If x_IδβJA, then xA.

    (ii) If xA, then x¯IδβJA.

    (iii) If x_IβJA, then xA.

    (iv) If xA, then x¯IβJA.

    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 aA, but a_IδβRA.

    (ii) A={a,b,c,d}, then e¯IδβRA, but eA.

    (iii) A={a,c,d,e}, then aA, but a_IβRA.

    (iv) A={b,c,d}, then e¯IβRA, but eA.

    Proposition 7.1. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) If x_JAx_ηJAx_IηJAx_IδβJA.

    (ii) If x¯IδβJAx¯IηJAx¯ηJAx¯JA.

    (iii) If x_JAx_ηJAx_IηJAx_IβJA.

    (iv) If x¯IβJAx¯IηJAx¯ηJAx¯JA.

    Proof. I prove (i) and the other similarly. x_JAx_ηJAx_IηJA by Proposition 2.6. Let x_IηJA. Then, xR_IηJ(A)xR_IδβJ(A)(byTheorem4.1)x_IδβJA.

    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_IδβRA, but b_βRA.

    (ii) Example 3.2 that if A={b,c}, then a¯IβRA, but a¯IδβRA.

    (iii) Example 3.3 that if A={e}, then e_IβRA, but e_βRA.

    (iv) Example 3.3 that if A={a,b,c,d}, then e¯IβRA, but e¯IβR.

    Definition 7.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X,AX and xX. The IδβJ-rough membership functions of A are defined by μIδβJA[0,1], where

    μIδβJA(x)={1if1ψIδβJA(x).min(ψIδβJA(x))otherwise.}.andψIδβJA(x)=|IδβJ(x)A||IδβJ(x)|,xIδβJ(x),IδβJ(x)IδβJO(X).

    Remark 7.4. The IδβJ-rough membership functions are used to define the I-δβJ-lower and I-δβJ-upper approximations as follows:

    (i) R_IδβJ(A)={xX:μIδβJA(x)=1}.

    (ii) ¯RIδβJ(A)={xX:μIδβJA(x)>0}.

    (iii) BNDIδβJ(A)={xX:0<μIδβJA(x)<1}.

    The following results give the fundamental properties of the IδβJ-rough membership functions.

    Proposition 7.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A,BX. Then

    (i) If μIδβJA(x)=1x_IδβJA.

    (ii) If μIδβJA(x)=0xX¯RIδβJ(A).

    (iii) If 0<μIδβJA(x)<1xBNDIδβJ(A).

    (iv) If μIδβJA(x)=1μIδβJA(x),xX.

    (v) If μIδβJAB(x)max(μIζJA(x),μIδβJB(x)),xX.

    (vi) If μIδβJAB(x)min(μIδβJA(x),μIδβJB(x)),xX.

    Proof. I prove (i), and the others similarly.

    x_IδβJAxR_IδβJ(A). Since R_IδβJ(A) is IδβJ-open set contained in A, thus |R_IδβJ(A)A||R_IδβJ(A)(A)|=|R_IδβJ(A)||R_IδβJ(A)|=1. Then, 1ψIδβJA(x) and accordingly μIδβJA(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 I [12] and I-δβJ-rough membership functions.

    Lemma 7.2. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) μJA(x)=1μηJA(x)=1μIηJA(x)=1μIδβJA(x)=1,xX.

    (ii) μJA(x)=0μηJA(x)=0μIηJA(x)=0μIδβJA(x)=0,xX.

    Proof. (i) μJA(x)=1μηJA(x)=1μIηJA(x)=1 directly from Lemma 2.1. Let μIηJA(x)=1, then xR_IηJ(A)xR_IδβJ(A)μIδβJA(x)=1,xX.

    (ii) μJA(x)=0μηJA(x)=0 directly from Lemma 2.1. Let μηJA(x)=0, then xX¯RIηJ(A)xX¯RIδβJ(A)μIδβJA(x)=0,xX.

    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,R,ΞJ) be a J-ndS, I be an ideal on X,AX and xX. The IβJ-rough membership functions of a J-ndS on X for a A are defines by μIβJA[0,1], where

    μIβJA(x)={1if1ψIβJA(x).min(ψIβJA(x))otherwise.}.andψIβJA(x)=|IβJ(x)A||IβJ(x)|,xIβJ(x),IβJ(x)IβJO(X).

    Remark 7.6. The I-J-nearly rough membership functions are used to define the I-βJ-lower and I-βJ-upper approximations as follows:

    (i) R_IβJ(A)={xX:μIβJA(x)=1}.

    (ii) ¯RIβJ(A)={xX:μIβJA(x)>0}.

    (iii) BNDIβJ(A)={xX:0<μIβJA(x)<1}.

    The following results give the fundamental properties of the IβJ-rough membership functions.

    Proposition 7.3. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and A,BX. Then

    (i) If μIβJA(x)=1x_IβJA.

    (ii) If μIβJA(x)=0xX¯RIβJ(A).

    (iii) If 0<μIβJA(x)<1xBNDIβJ(A).

    (iv) If μIβJA(x)=1μIβJA(x),xX.

    (v) If μIβJAB(x)max(μIβJA(x),μIβJB(x)),xX.

    (vi) If μIβJAB(x)min(μIβJA(x),μIβJB(x)),xX.

    Proof. It is similar to Proposition 7.2.

    Lemma 7.3. Let (X,R,ΞJ) be a J-ndS, I be an ideal on X and AX. Then

    (i) μJA(x)=1μηJA(x)=1μIηJA(x)=1μIβJA(x)=1,xX.

    (ii) μJA(x)=0μηJA(x)=0μIηJA(x)=0μIβJA(x)=0,xX.

    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={x1,x2,x3,x4,x5} be five amino acids (for short, AAs). The (AAs) are described in terms of seven attributes: a1= PIE and a2= SAC = surface area, a3= MR = molecular refractivity, a4= LAM = the side chain polarity and a5= Vol = molecular volume ([10,34]). Table 3 shows all quantitative attributes of five AAs.

    Table 3.  Quantitative attributes of five amino acids.
    a1 a2 a3 a4 a5
    {x1} 0.23 254.2 2.216 0.02 82.2
    {x2} 0.48 303.6 2.994 1.24 112.3
    {x3} 0.61 287.9 2.994 1.08 103.7
    {x4} 0.45 282.9 2.933 0.11 99.1
    {x5} 0.11 335.0 3.458 0.19 127.5

     | Show Table
    DownLoad: CSV

    I consider the relations on X defined as: Ri={(xi,xj):xi(ak)xj(ak)<σk2,i,j,k=1,2,...,5} where σk represents the standard deviation of the quantitative attributes.

    The right neighborhoods xX with respect to the relations are shown in Table 4.

    Table 4.  Right neighborhood of seven reflexive relations.
    $$ xiR1 xiR2 xiR3 xiR4 xiR5
    {x1} {x1,x4} X X {x1,x4,x5} X
    {x2} X {x2,x5} {x2,x3,x4,x5} X {x2,x5}
    {x3} X {x2,x3,x4,x5} {x2,x3,x4,x5} X {x2,x3,x4,x5}
    {x4} {x4} {x2,x3,x4,x5} {x2,x3,x4,x5} {x1,x4,x5} {x2,x3,x4,x5}
    {x5} {x1,x4,x5} {x5} {x5} {x1,x4,x5} {x3,x5}

     | Show Table
    DownLoad: CSV

    The intersection of all right neighborhoods xX is:

    x1R=5k=1(x1Rk)={x1,x4},x2R=5k=1(x2Rk)={x2,x5},x3R=5k=1(x3Rk)={x2,x3,x4,x5},x4R=5k=1(x4Rk)={x4},x5R=5k=1(x5Rk)={x5}. Then, τR={ϕ,X,{x4},{x5},{x4,x5},{x1,x4},{x2,x5},{x1,x4,x5},{x2,x4,x5},{x1,x2,x4,x5},{x1,x3,x4,x5}},βRO(X)={ϕ,X,{x4},{x5},{x1,x4},{x1,x5},{x2,x5},{x3,x4},{x3,x5},{x4,x5},{x1,x3,x5},{x1,x4,x5},{x2,x3,x5},{x2,x4,x5},,{x3,x4,x5},{x1,x2,x4,x5},{x1,x3,x4,x5},{x2,x3,x4,x5}}. Let I={ϕ,{x1}}, then I-δβRO(X)=P(X).

    (i) It is clear that every βR-open is I-δβR-open, but the converse is not necessary to be true. For example take A={x1} which is I-δβR-open, but it is not βR-open. Hence, the current concept generalize and extend the previous one 2.6 in [8].

    (ii) The current approximations which are depended on I-δβR-open is better than the previous approximations 2.7 [8] which depended on βR-open. As for any concept AX (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 I-δβR-open in general is greater than the accuracy measure by using βR-open. For example take A={x1,x4,x5}, Then,

    (a) by the current Definition 4.1, BNDIδβR(A)=ϕ and ACCIδβR(A)=1;

    (b) by the previous one in Definition 2.7 [8], BNDβR(A)={2,3} and ACCβR(A)=35.

    (iii) Similarly, it is easy to calculate I-βRO(X),τIβR 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:

    Table 5.  Decision system.
    Student Science German Mathematics Decision
    {x1} Bad Good Medium Accept
    {x2} Good Bad Medium Accept
    {x3} Good Good Good Accept
    {x4} Bad Good Bad Reject
    {x5} Good Bad Medium Reject
    {x6} Bad Good Good Accept

     | Show Table
    DownLoad: CSV

    From Table 5:

    (i) The set of universe: X={x1,x2,x3,x4,x5,x6}.

    (ii) The set of attributes: AT ={Science,German,Mathematics}.

    (iii) The sets of values:

    VScience={Bad,Good},VGerman={Bad,Good},VMathematics={Bad,Medium,Good}

    and

    VDecision={Accept,Reject}.

    I take the set of condition attributes, C={Science,German,Mathematics}. Thus, the corresponding equivalence relation is R={(x1,x1),(x2,x5),(x3,x3),(x4,x4),(x5,x2),(x6,x6)}, let I={ϕ,{x1}}. Then, I-δβRO(X)=P(X). Let A (Decision: Accept) ={x1,x2,x3,x6}. Then

    (i) The rough membership functions with respect to the Definition of Pawlak and Skowron 19 [28] are computed as follows:

    μA(x1)=μA(x3)=μA(x6)=1,μA(x2)=12.

    (ii) The I-δβR-rough membership functions in Definition 7.2 are calculated as follows:

    μIδβRA(x1)=μIδβRA(x2)=μIδβRA(x3)=μIδβRA(x6)=1.

    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 x5

    (i) The rough membership function with respect to the Definition of Pawlak and Skowron 2.19 [28] is μA(x5)=12. This means that x5 may belong to the set A (Decision: Accept), A={x1,x2,x3,x6} and this contradicts to Table 5.

    (ii) The I-δβR-rough membership function in Definition 7.2 is μIδβRA(x5)=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.



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