Thư viện số Văn Lang: Applied Computing and Informatics: Volume 12, Issue 1

ORIGINAL ARTICLE

Generalized covering approximation space and near concepts with some applications

M.E. Abd El-Monsef, A.M. Kozae, M.K. El-Bably

^*

Department of Mathematics, Faculty of Science, Tanta University, Egypt

Received 27 May 2014; revised 10 February 2015; accepted 11 February 2015 Available online 23 March 2015

KEYWORDS Coverings;

Generalized covering approximation space;

Near concepts;

Memberships rela- tions and functions;

Fuzzy sets

Abstract In this paper, we shall integrate some ideas in terms of concepts in topology. First, we introduce some new concepts of rough membership relations and functions in the generalized covering approximation space. Second, we introduce some topological applications namely ‘‘near concepts’’ in the general- ized covering approximation space. Accordingly, several types of fuzzy sets are constructed. The basic notions of near approximations are introduced and suffi- ciently illustrated. Near concepts are provided to be easy tools to classify the sets and to help for measuring exactness and roughness of sets. Many proved results, examples and counter examples are provided. Finally, we give two practical examples to illustrate our approaches.

ª2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/).

* Corresponding author.

E-mail addresses:monsef@dr.com(M.E. Abd El-Monsef),akozae55@yahoo.com(A.M. Kozae),mkamel_ba- bly@yahoo.com(M.K. El-Bably).

Peer review under responsibility of King Saud University.

Production and hosting by Elsevier

Saudi Computer Society, King Saud University

Applied Computing and Informatics

(http://computer.org.sa) www.ksu.edu.sa www.sciencedirect.com

http://dx.doi.org/10.1016/j.aci.2015.02.001

2210-8327ª2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets[1–9,11,12]is a natural extension of clas- sical rough sets by relaxing the partitions arising from equivalence relations to cov- erings. In our work[6], we have introduced a framework to generalize covering approximation space that was introduced by Zhu[11]. In fact, we have introduced the generalized covering approximation spaceGn–CASas a generalization to rough set theory and covering approximation space. TheGn–CASis defined by the triple hU;R;Cni, whereU–;be a finite set,Rbe a binary relation onUandCnben-cover ofUassociated toR, wheren2 fr;lg(for more details see[6]).

The main works in this paper are divided into three parts. In the beginning of work, we introduce some new generalized definitions to rough membership rela- tions (resp. functions) and new types of fuzzy sets inG_n–CAS. Second part aims to introduce one of an important topological concepts which are called ‘‘near con- cepts’’ in rough context (specially, inGn–CAS). In fact, we apply near concepts in Gn –CASto define different tools for modifying the original operations. The sug- gested methods in this paper represent easy mathematical tools to approximate the rough sets and removing the uncertainty (vagueness) of sets. In addition, compar- isons between the suggested methods are obtained and many examples (resp.

counter examples) to illustrate these connections are provided. Hence, we can say that our approaches are very useful in rough context namely, in information analysis and in decision making. Finally, in the end of paper, simple practical examples are provided to illustrate the suggested methods and to show the impor- tance of these methods in rough context namely in information system and in multi-valued information system. In addition, we give some comparisons between our approaches and others approaches such as Pawlak and Lin approaches.

2.j-Rough membership relations, j-rough membership functions andj-fuzzy sets The present section is devoted to introduce new definitions for rough membership relations and functions as easy tools to classify the sets and help for measuring exactness and roughness of sets. These rough membership functions allow us to define four different fuzzy sets inGn–CAS. Moreover, the suggested rough mem- bership relations (resp. functions) are more accurate than classical rough member- ship function that was given by Lin[10] and the other types.

Definition 2.1. LethU;R;Cni be a Gn–CAS, andA#U. Then we say that:

(i) xis ‘‘j-surely’’ belongs toA, writtenx2jA, ifx2 Rjð Þ.A (ii) xis ‘‘j-possibly’’ belongs toX, writtenx2jA, ifx2 RjðAÞ.

These two rough membership relations are called ‘‘j-strong’’ and ‘‘j-weak’’

membership relations respectively.

Lemma 2.1. Let hU;R;Cni be a Gn –CAS, and A#U. Then the following state- ments are true in general:

(i)x2jAimpliesx2A. (ii)x2Aimpliesx2jA.

Proof. Straightforward.

The converse of the above lemma is not true in general, as the following exam- ple illustrates:

Example 2.1. Let hU;R;Cni be a Gn–CAS, where U¼fa;b;c;dg and R ¼fða;aÞ; ðb;bÞ; ðc;cÞ; ðc;bÞ; ðc;dÞ;ðd;aÞg. We will show the above remark in case ofj¼rand the other cases similarly: Suppose thatA¼fa;b;dg, then we get Rrð Þ ¼A fa;bg and RrðAÞ ¼U. Clearlyd2AbutdRrAand c2rA butc R A.

The following proposition is very interesting since it is give the relationships between different types of membership relations 2j and 2j. Accordingly, we will illustrate the importance of using these different types of these membership relations.

Proposition 2.1. Let hU;R;Cni be a Gn–CAS, and A#U. Then the following are true generally

(i)If x2iA ) x2rA ) x2uA. (iii)If x2uA ) x2rA) x2iA.

(ii)If x2iA ) x2lA ) x2uA. (iv)If x2uA) x2lA ) x2iA.

Proof. We will prove the first statement and the others similarly:

ð Þi Ifx2iA ) x2 Rið Þ )A x2 Rrð Þ )A x2r A.

Also, if x2rA ) x2 Rrð Þ )A x2 Ruð Þ )A x2uA.

The converse of the above proposition is not true in general as the following example illustrates.

Example 2.2. Let hU;R;Cni be a Gn–CAS, where U¼fa;b;c;dg and R ¼fða;aÞ; ða;bÞ; ðb;cÞ; ðb;dÞ; ðc;aÞ; ðd;aÞg. Suppose that A¼fb;c;dg. Then Ruð Þ ¼ ;,A Rrð Þ ¼A fc;dg, Rlð Þ ¼A f gb and Rið Þ ¼A fb;c;dg. Accordingly, c2rA and b2lA but bRuA and cRuA. Also b2iA and c2iA but bRrA and cRlA. By similar way, we can illustrate the others cases.

Definition 2.2. Let hU;R;Cni be a Gn–CAS, and A#U. Then for each j2fr;l;i;ug and x2U:

Thej- rough membership functions onUfor subsetAarel_A^j :U!½0;1where l_A^jð Þ ¼x j^{Njð Þ\Ax} j

Njð Þx

j j and j jA denotes the cardinality of A.

The rough j-membership function expresses conditional probability that x belongs toAgivenRand can be interpreted as a degree thatxbelongs toAin view of information aboutxexpressed byR. Moreover, in case of infinite universe, the above membership functionl_A^j can be use for spaces having locally finite minimal neighborhoods for each point.

Lemma 2.2. LethU;R;C_ni be aG_n–CAS, and A#U. Then for every x2U

(i)l^u_Að Þ ¼x 1 ) l^r_Að Þ ¼x 1 ) lⁱ_Að Þ ¼x 1. (ii)l^u_Að Þ ¼x 0 ) l^r_Að Þ ¼x 0 ) lⁱ_Að Þ ¼x 0.

(iii)l^u_Að Þ ¼x 1 ) l^l_Að Þ ¼x 1 ) lⁱ_Að Þ ¼x 1. (iv)l^u_Að Þ ¼x 0 ) l^l_Að Þ ¼x 0 ) lⁱ_Að Þ ¼x 0.

Proof. We will prove first statement and the others similarly:

(i) l^u_Að Þ ¼x 1 ) x2uA ) x2rA ) l^r_Að Þ ¼x 1. Also, l^r_Að Þ ¼x 1 ) x2r A ) x2iA ) lⁱ_Að Þ ¼x 1:

Remark 2.1.

(i) According to the above results, we can prove that lⁱ_A is more accurate than the others types that is:

(1) If x2A ) l^u_Að Þx 6l^r_Að Þx 6lⁱ_Að Þx and ifx2A ) l^u_Að Þx 6l^l_Að Þx 6lⁱ_Að Þ.x

(2) If x R A ) lⁱ_Að Þx 6l^r_Að Þx 6l^u_Að Þx and ifx2A ) lⁱ_Að Þx 6l^l_Að Þx 6l^u_Að Þ.x

(ii) The converse of the above lemma is not true in general.

The following example illustrates Remark 2.1.

Example 2.3. According toExample 2.2, consider the subsetA¼fb;c;dg. Then we get

l^r_Að Þ ¼a ^jfag\A_jfagj^j¼0. l^l_Að Þ ¼a ^jfag\A_jfagj^j¼0.

l^r_Að Þ ¼b ^jfa;bg\A_jfa;bgj^j¼¹₂. l^l_Að Þ ¼b ^jfbg\A_jfbgj^j¼1.

l^r_Að Þ ¼c ^jfc;dg\A_jfc;dj ^j¼1. l^l_Að Þ ¼c ^jfa;c;dg\A_jfa;c;dgj^j¼²₃. l^r_Að Þ ¼d ^jfc;dg\A_jfc;dgj^j¼1. l^l_Að Þ ¼d ^jfa;c;dg\A_jfa;c;dgj^j¼²₃.

lⁱ_Að Þ ¼a ^jfag\A_jfagj^j¼0. l^u_Að Þ ¼a ^jfag\A_jfagj^j¼0.

lⁱ_Að Þ ¼b ^jfbg\A_jfbgj^j¼1. l^u_Að Þ ¼b ^jfa;bg\A_jfa;bgj^j¼¹₂. lⁱ_Að Þ ¼c ^jfc;dg\A_jfc;dgj^j¼1. l^u_Að Þ ¼c ^jfa;c;dg\A_jfa;c;dgj^j¼²₃. lⁱ_Að Þ ¼d ^jfc;dg\A_jfc;dgj^j¼1. l^u_Að Þ ¼d ^jfa;c;dg\A_jfa;c;dgj^j¼²₃.

One of the key issues in all fuzzy sets is how to determine fuzzy membership functions. A membership functions provides a measure of the degree of similarity of element to fuzzy set. The following definition uses the j-rough membership functions l_A^j to define four different types of fuzzy sets inGn–CAS.

Definition 2.3. LethU;R;Cnibe aGn–CAS, andA#U. Then thej-fuzzy sets inU is the set of ordered pairs:

Ae_j ¼x;l_A^jð Þx x2U

; 8j2 fr;l;i;ug:

Example 2.4. According to Example 2.3, consider the subsetA¼fb;c;dg. Then we get

Aer ¼ ða;0Þ; b;1 2

;ðc;1Þ;ðd;1Þ

; Ael¼ ða;0Þ;ðb;1Þ; c;2 3

; d;2

3

;

Ae_u ¼ ða;0Þ; b;1 2

; c;2

3

; d;2

3

; and Ae_i¼fða;0Þ;ðb;1Þ;ðc;1Þ;ðd;1Þg:

3. Near rough concepts in the generalized covering approximation spaceG_n CAS The main goal of this section is to introduce one of the important topological applications which are named ‘‘near concepts’’ inGn–CAS. Moreover, we intro- duce the new concepts ‘‘j-near approximations’’ (resp. j-near boundary regions and j-near accuracy measures) to generalize the j-approximations (resp. j- boundary regions andj-accuracy measures). In addition, we introduce near exact- ness and near roughness by applying near concepts to make more accuracy for definability of sets in Gn–CAS.

Definition 3.1. Let hU;R;Cni be a Gn–CAS, and A#U. Thus we define ‘‘near rough’’ sets in Uas follows: For each j2fr;l;i;ug, the subsetA is called

(i) j-Pre rough set (briefly p_jÞif A#R_jR_jðAÞ . (ii) j-Semi rough set (brieflysjÞif A#RjRjðAÞ

. (iii) cj-rough set if A# Rj Rj

[ RjRjðAÞ

.

The above sets are called ‘‘j-near rough sets’’ and the families of j-near rough sets of Udenotes by Kjð Þ, for eachU K2 fP;S;cg.

Remark 3.1. The family ofj-pre rough sets and the family ofj-semi rough sets are not comparable as the following example illustrates.

Example 3.1. LethU;R;Cni be aGn–CAS, where U¼fa;b;c;dg and R ¼fða;aÞ;ða;bÞ;ðb;aÞ;ðb;bÞ;ðc;aÞ;ðc;bÞ;ðc;cÞ;ðc;dÞ;ðd;dÞg:

We will show the above remark in case of j¼r and the other cases similarly as follows:

Nrð Þ ¼a fa;bg ¼Nrð Þ;b Nrð Þ ¼c U; Nrð Þ ¼d f g:d

Thus, we compute the j-near rough sets forj¼ras follows:

The family of r-pre rough sets is: Prð Þ ¼U fU;;;f g;a f g;b f g;d fa;bg;fa;dg;

b;d

f g;fa;b;dg;fa;c;dg;fb;c;dgg.

The family of r-semi rough sets is: Srð Þ ¼U fU;;;f g;d fa;bg;fc;dg;

a;b;c

f g;fa;b;dgg.

The main goal of the following definitions is to introduce the new approxima- tion operators (j-near approximations) which modify and generalize the j- approximations.

Definition 3.2. LethU;R;Cnibe aGn–CAS, andA#U. Then we define thej-near approximations of any subset Aas follows: For eachj2fr;l;i;ug

(i) Thej-pre lower and the j-pre upper approximations of Aare defined respec- tively by

R^p_jð Þ ¼A x2AjNjð Þx #RjðAÞ

and R^p_jðAÞ ¼A[ fx2A^cjNjðxÞ \ RjðAÞ–;g (ii) The j-semi lower and the j-semi upper approximations of A are defined

respectively by

R^s_jðAÞ ¼x2A N jðxÞ \ RjðAÞ–;

and R^s_jðAÞ ¼A[x2 RjðAÞNjðxÞ#RjðAÞ (iii) Thec_j-lower and thec_j-upper approximations ofAare defined respectively by R^c_jðAÞ ¼ R^p_jðAÞ [ R^s_jðAÞ and R^c_jðAÞ ¼ R^p_jðAÞ \ R^s_jðAÞ

Definition 3.3. LethU;R;Cnibe aGn–CAS, andA#U. Then we define thej-near boundary,j-near positive andj-near negative regions ofAare defined respectively as follows:

8j2fr;l;i;ug; k2 fp;s;cg:B^k_jðAÞ ¼ R^k_jðAÞ R^k_jðAÞ; POS^k_jð ÞA

¼ R^k_jð ÞA and NEG^k_jðAÞ ¼U R^k_jðAÞ:

Definition 3.4. Let hU;R;Cni be a Gn–CAS, and A#U. Then, for each j2fr;l;i;ug; k2 fp;s;cg, the j-near accuracy of the j-near approximations of A#Uis defined by

d^k_jðAÞ ¼

R^k_jðAÞ

R^k_jðAÞ

; whereR^k_jðAÞ–0: Obviously 06d^k_jð ÞA 61:

Definition 3.5. Let hU;R;Cni be a Gn–CAS, and A#U. Then, for each j2fr;l;i;ug; k2 fp;s;cg, the subset A is called ‘‘j-near definable (briefly kj- exact) set’’ if R^k_jð Þ ¼ RA ^k_jðAÞ ¼A. Otherwise, it is called j-near rough (briefly k_j-rough). It is clear that A is k_j-exact if d^k_jð Þ ¼A 1 and B^k_jð Þ ¼ ;. Otherwise, itA isk_j-rough.

Remark 3.2. In the G_n–CAS; hU;R;C_ni, we can compute thej-near approxima- tions of any subsetA#U, directly by using thej-approximations, as the following lemma illustrates.

Lemma 3.1. LethU;R;C_nibe aG_n–CAS, and A#U. Then, for each j2fr;l;i;ug:

(i)R^p_jðAÞ ¼A\ RjRjðAÞ

(ii)R^p_jðAÞ ¼A[ RjRjðAÞ (iii)R^s_jðAÞ ¼A\ RjRjðAÞ

(iv)R^s_jðAÞ ¼A[ RjRjðAÞ

Proof. FromDefinition 3.2, the proof is obvious.

Lemma 3.2. Let hU;R;C_ni be a G_n–CAS, and A#U. Then, for each j2fr;l;i;ug; k2 fp;s;cg:

The subset A is kj-rough set if A¼ R^k_jðAÞ.

The following proposition introduces the fundamental properties of the j-near approximations.

Proposition 3.1. Let hU;R;Cni be a Gn–CAS, and A;B#U. Then, 8j2fr;l;i;ug; k2 fp;s;cg:

(1)R^k_jð ÞA #A#R^k_jðAÞ (7)R^k_jðA[BÞ R^k_jð Þ [ RA ^k_jð Þ.B (2)R^k_jð Þ ¼ RU ^k_jðUÞ ¼U and (8)R^k_jðA[BÞ R^k_jðAÞ [ R^k_jðBÞ:

R^k_jð;Þ ¼ R^k_jð;Þ ¼ ;: (9)R^k_jð Þ ¼ RA h ^k_jðA^cÞic

andR^k_jðAÞ ¼ Rh ^k_jðA^cÞic

; (3)If A#B thenR^k_jð ÞA #R^k_jð Þ.B where A^cis the complement of A.

(4)If A#B thenR^k_jðAÞ#R^k_jðBÞ. (10)R^k_jR^k_jð ÞA

¼ R^k_jð Þ.A (5)R^k_jðA\BÞ#R^k_jð Þ \ RA ^k_jð Þ:B (11)R^k_jR^k_jðAÞ

¼ R^k_jðAÞ:

(6)R^k_jðA\BÞ#R^k_jðAÞ \ R^k_jðBÞ:

Proof. Firstly, the proof of (1), (2), (10) and (11) is obvious directly from Definition 3.2.

Now, we will prove the left properties for case k¼p and the other cases similarly.

(3) If A#B then R^p_jðAÞ ¼x2AjNjðxÞ#RjðAÞ

#x2BjNjðxÞ#RjðBÞ

¼ R^p_jðBÞ.

The proof of (4)–(8), by similar way as 3ð Þ.

ð Þ9 From Lemma 3.1, we get R^p_jðA^cÞ

c

¼A\ R_jR_jðAÞc

¼A^c[ R _jR_jðAÞc

¼A^c[ R_jR_jðAÞ

¼ R^p_jðAÞ:

Similarly,R^p_jðAÞ ¼ R ^p_jðA^cÞc

. h

The following results introduce the relationships between thej-approximations and the j-near approximations. Moreover, these results show the importance of applying near concepts inGn–CAS.

Theorem 3.1. Let hU;R;Cni be a Gn–CAS, and A#U. Then, 8j2fr;l;i;ug; k2 fp;s;cg :

RjðAÞ#R^k_jðAÞ#A#R^k_jðAÞ#RjðAÞ

Proof. We will prove the proposition in case of k¼p and the other cases similarly:

Let x2 Rjð Þ, thenA x2A such that NjðxÞ#A. Thus x2A such that NjðxÞ#RjðAÞ and this impliesx2 R^p_jð Þ. By duality, we getA R^p_jðAÞ#RjðAÞ.

Corollary 3.1. Let hU;R;C_ni be a G_n–CAS, and A#U. Then 8j2fr;l;i;ug; k2 fp;s;cg:

(1)B^k_jð ÞA #B_jð Þ:A (2)djð ÞA 6d^k_jð Þ:A

Remark 3.3. The main goals of the following example are:

(i) The converse of the above results is not true in general.

(ii) Using near concepts in rough context is very useful for removing the vague- ness of sets and accordingly, these approaches is very useful in decision making.

Example 3.2. LethU;R;Cni be aGn–CAS, whereU¼fa;b;c;dg and R ¼fða;aÞ;ða;bÞ;ðb;aÞ;ðb;bÞ;ðc;aÞ;ðc;bÞ;ðc;cÞ;ðc;dÞ;ðd;dÞg:

Thus, we getNrð Þ ¼a fa;bg ¼Nrð Þ,b Nrð Þ ¼c U, Nrð Þ ¼d f g.d

By using Definitions 3.2 and 3.4, the following table gives comparisons between thej-accuracy of approximations and thekj-accuracy of approximations of the all subsets ofU, where j¼r;8k2 fp;s;cg:

From Table 3.1, we notice that:

(i) Using c_r in constructing the approximations of sets is more accurate than others types, since for any subset A#U; d_rð ÞA 6d^c_rð ÞA and d^k_rð ÞA 6d^c_rð Þ;A 8k 2 fp;sg. Thus, these approaches will helps to extract and discovery the hidden information in data that collected from real-life appli- cations. For example, all shaded sets in Table 3.1.

(ii) Every r-exact set is r-near exact, but the converse is not true. For example, shaded sets inTable 3.1.

Remark 3.4. The following result is very interesting because it is prove that thej- near approaches are more accurate than the j-approaches. Moreover, it is illus- trates the importance of j-near concepts in exactness of sets.

Proposition 3.2. Let hU;R;Cni be a Gn–CAS, and A#U. Then 8j2fr;l;i;ug; k2 fp;s;cg, the following is true in general: If A is j-exact set, then it is kj-exact.

Table 3.1 Comparisons between the j-accuracy and the kj-accuracy of approximations of the all subsets ofU.

Proof. IfAisj-exact set, thenB_jð Þ ¼ ;. Thus, by Corollary 3.1,A B^k_jð Þ ¼ ;A and accordingly Aisk_j-exact.

The converse of the above proposition is not true in general as Example 3.2 illustrates.

The main goal of the following results is to introduce the relationships between different types ofj-near approximations, j-near boundary, j-near accuracy and j- near exactness respectively.

Proposition 3.3. Let hU;R;Cni be Gn–CAS and A#U. Then, 8j2fr;l;i;ug, the following statements are true in general:

(i)R^p_jðAÞ#R^c_jðAÞ. (iii)R^c_jðAÞ#R^p_jðAÞ:

(ii)R^s_jðAÞ#R^c_jðAÞ. (iv)R^c_jðAÞ#R^s_jðAÞ:

Proof. By using Lemmas 3.1 and 3.2, the proof is obvious.

Corollary 3.2. LethU;R;Cni be a Gn –CAS, and A#U. Then8j2fr;l;i;ug, the following statements are true in general:

(i)d^p_jðAÞ6d^c_jðAÞ. (iii)B^c_jðAÞ#B^p_jðAÞ.

(ii)d^S_jðAÞ6d^c_jðAÞ. (iv)B^c_jðAÞ#B^s_jðAÞ.

Corollary 3.3. LethU;R;C_ni be a G_n –CAS, and A#U. Then8j2fr;l;i;ug, the following statements are true in general:

(i)A is p_j-exact ) A isc_j-exact: (ii)A is sj-exact ) A isc_j-exact:

Remark 3.5.

(i) The converse of the above results is not true in general as the following exam- ple illustrates.

(ii) 8j2fr;l;i;ug; d^c_jð Þ ¼A maxd^p_jð Þ;A d^S_jð ÞA

, wheremaxrepresents the maxi- mum of two quantities.

Example 3.3. According toExample 3.2, it is clear thatAisc_r-exact but it is nots_r- exact. In addition, the subset Bis c_r-exact but it is notp_r-exact.

The relationships between different types of j-near approximations (for each j2fr;l;i;ugÞare not comparable (no it is not like to thej-approximations as in[6]) as the following remark illustrates.

Remark 3.6. LethU;R;Cnibe aGn –CAS, andA#U. Then8k2 fp;s;cg, the fol- lowing statements are not true in general:

(i)R^k_uð ÞA #R^k_rð ÞA #R^k_ið Þ.A (iii)R^k_iðAÞ#R^k_rðAÞ#R^k_uðAÞ:

(ii)R^k_uð ÞA #R^k_lð Þ#A R^k_ið Þ.A (iv)R^k_iðAÞ#R^k_lðAÞ#R^k_uðAÞ:

The following example illustrates this remark.

Example 3.4. LethU;R;C_ni be aG_n–CAS, whereU¼fa;b;c;dg and R ¼fða;aÞ;ðb;bÞ;ðb;aÞ;ðc;aÞ;ðc;dÞ;ðd;aÞ;ðd;cÞ;ðd;aÞg:

Now suppose that A¼f g;b B¼fa;dg and C¼fa;cg. Thus, we get R^p_rð Þ ¼ ;,A but R^p_uð Þ ¼A A and R^p_lð Þ ¼B f g, butd R^p_uð Þ ¼ fa;B dg. In addition, we have R^s_ið Þ ¼ fag, butC R^s_rð Þ ¼C A.

4.j-near rough membership relations, j-near rough membership functions andj-near fuzzy sets in Gn CAS

By considering j-near concepts, the new concepts ‘‘j-near rough membership rela- tions’’ (resp. ‘‘j-near rough membership functions’’) are provided to modify and generalize the j-membership relations (resp. j-membership functions) in G_n–CAS. The near rough membership functions are considered as easy tools to classify the sets and help for measuring near exactness and near roughness of sets.

The existence of near rough membership functions made us introduce the concept of near fuzzy sets.

Definition 4.1. Let hU;R;Cni be a Gn–CAS, and A#U. Then 8j2fr;l;i;ug;k2 fp;s;cg, we say:

(i) x is ‘‘j-near surely’’ (briefly kj-surely) belongs to A, written x2^k_j A, if x2 R^k_jð Þ.A

(ii) x is ‘‘j-near possibly’’ (briefly kj-possibly) belongs to X, written x2^k_j A, if x2 R^k_jðAÞ.

Lemma 4.1. Let hU;R;Cni be a Gn–CAS, and A#U. Thus 8j2fr;l;i;ug; k2 fp;s;cg, the following statements are true in general:

(i)If x2^k_j A implies to x2A. (ii)If x2A implies to x2jA.

Proof. Straightforward.

The converse of the above lemma is not true in general, as the following exam- ple illustrates:

Example 4.1. Consider U¼fa;b;c;dg and R ¼fða;aÞ;ðb;bÞ;ðb;aÞ;ðc;aÞ;ðc;dÞ;

d;a

ð Þ;ðd;cÞ;ðd;aÞg.

Thus, we getNrð Þ ¼a f g;a Nrð Þ ¼b fa;bg, Nrð Þ ¼c fa;c;dg; Nrð Þ ¼d f g.d We will show the above remark in case ofðj¼r and k¼pÞand the other cases similarly:

Suppose thatA¼fb;dg, then we getR^p_rð Þ ¼A f gd and R^p_rðAÞ ¼ fb;c;dg.

Clearlyd2AbutdR^p_r A andc2^p_r Abutc R A.

Remark 4.1. We can redefine the j-near approximations by using 2^k_j and 2^k_j as follows:

For anyA;B#U: R^k_jð Þ ¼A nx2U x 2^k_j Ao

andR^k_jðAÞ ¼nx2U x 2^k_j Ao . The following proposition is very interesting since it is give the relationships between thej-rough membership relations andj-near rough membership relations.

Accordingly, we will show the importance of using these different types ofj-near rough membership relations.

Proposition 4.1. Let hU;R;Cni be a Gn–CAS, and A#U. Then 8j2fr;l;i;ug; k2 fp;s;cg, the following statements are true in general:

(i)x2jA ) x2^k_j A. (ii)x2^k_j A ) x2jA.

Proof. We will prove first statement and the other similarly:

ðiÞ x2jA ) x2 RjðAÞ ) x2 R^k_jðAÞ ) x2^k_j A:

The converse of the above proposition is not true in general as the following example illustrates.

Example 4.2. Let U¼fa;b;c;dg and

R ¼fða;aÞ;ðb;bÞ;ðb;aÞ;ðc;aÞ;ðc;dÞ;ðd;aÞ;ðd;cÞ;ðd;aÞg.

We will show the above remark in case ofðj¼r and k¼sÞand the other cases similarly:

Suppose that A¼fa;cg and B¼ fb;dg, then we get Rrð Þ ¼A f ga and R^s_rð Þ ¼ fa;A cg.

Clearlyc2^s_rA, butcRrA althoughc2A.

Also RrðBÞ ¼ fb;c;dg and R^s_rðBÞ ¼ fb;dg. Clearly cR^s_rB; but c2rBalthough c R B.

Definition 4.2. LethU;R;Cnibe aGn–CAS, andA#U. Thus we can define thej- near rough membership functions for G_n–CAS as follows: For each j2fr;l;i;ug; k2 fp;s;cg and x2U, the j-near rough membership functions on Ufor subsetA arel^k_A^j :U! ½0;1, where

l^k_A^jð Þ ¼x

1 if 12W^k_A^jð Þ:x min W^k_A^jð Þx

Otherwise:

8<

:

and W^k_A^jð Þ ¼x j^{kjð Þ\Ax} j

kjð Þx

j j such that k_jð Þx is a j-near rough set in that containsx.

The following result is very interesting since it gives the relation between the roughj-membership functions andj-near rough membership functions. Moreover, it illustrates the importance of j-near rough membership functions.

Lemma 4.2. Let hU;R;Cni be a Gn–CAS, and A;B#U. Then, for each j2fr;l;i;ugand k2 fp;s;cg, the following is true in general:

(i)l_A^jð Þ ¼x 1 ) l^k_A^jð Þ ¼x 1; 8x2U. (ii)l_A^jð Þ ¼x 0 ) l^k_A^jð Þ ¼x 0; 8x2U.

Proof. (i) Ifl_A^jð Þ ¼x 1, thenNjð Þx #A; 8x2U. Thusx2 Rjð ÞA and this implies x2 R^k_jð ÞA which is a j-near rough set contained in A. Accordingly, l^k_A^jð Þ ¼x 1; 8x2U.

(ii) Ifl_A^jð Þ ¼x 0, then Njð Þ \x A¼ ;; 8x2U. ButNjð Þx is a j-near rough set that containsx. h

Accordingly 02W^k_A^jð Þx and this means that minW^k_A^jð Þx

¼0. Hence l^k_A^jð Þ ¼x 0; 8x2U:

Remark 4.2.

(i) According to the above results, we can prove that l^k_A^j is more accurate than l_A^j, this means that:

ð1Þ If x2A ) l_A^jð Þx 6l^k_A^jð Þ:x ð2Þ If x R A ) l^k_A^jð Þx 6l_A^jð Þ:x (ii) The converse of Lemma 4.2 is not true in general.

The following example illustratesRemarks 4.2.

Example 4.3. LethU;R;Cni be aGn–CAS, where U¼fa;b;c;dg and R ¼fða;aÞ;ða;bÞ;ðb;aÞ;ðb;bÞ;ðc;aÞ;ðc;bÞ;ðc;cÞ;ðc;dÞ;ðd;dÞg:

We will show the above result in case ofj ¼randk¼sthe other cases similarly as follows:

First we haveNrð Þ ¼a fa;bg ¼Nrð Þ,b Nrð Þ ¼c U,Nrð Þ ¼d f g. Thus we can get.d

The family of all r-semi rough sets is:

Srð Þ ¼U fU;;;f g;a f g;d fa;bg;fa;dg;fa;cg;fc;dg;fa;b;cg;fa;b;dg;fa;c;dgg.

Now consider the subsetA¼ fa;cg, then ther-rough membership functions of A;x2Uare

l^r_Að Þ ¼a ^jfag\A_jfagj^j¼1. l^r_Að Þ ¼c ^jfa;c;dg\A_jfa;c;dj ^j¼²₃. l^r_Að Þ ¼b ^jfa;bg\A_jfa;bgj^j¼¹₂. l^r_Að Þ ¼d ^jfdg\A_jfdgj^j¼0.

But ther-semi rough membership functions of A;x2Uare W^s_A^rð Þ ¼a jfag \Aj

fag

j j ¼1; jfa;bg \Aj fa;bg

j j ¼1

2;. . .

) l^s_A^rð Þ ¼a 1:

W^s_A^rð Þ ¼b jfa;bg \Aj fa;bg

j j ¼1

2; jfa;b;cg \Aj fa;b;cg

j j ¼2

3; jfa;b;dg \Aj fa;b;dg

j j ¼1

3

) l^s_A^rð Þb

¼1 3:

W^s_A^rð Þ ¼c jfa;cg \Aj fa;cg

j j ¼1; jfc;dg \Aj fc;dg

j j ¼1

2;. . .

) l^s_A^rð Þ ¼c 1:

W^s_A^rð Þ ¼d jfdg \Aj fdg

j j ¼0; jfa;dg \Aj fa;dg

j j ¼1

2;. . .

) l^s_A^rð Þ ¼a 0:

The j-near rough membership functions l^k_A^j allow us to define twelve different types of fuzzy sets inG_n–CASas the following definition illustrates.

Definition 4.3. Let hU;R;Cni be a Gn–CAS, and A#U. Then for each j2fr;l;i;ug and k2 fp;s;cg, the j-near fuzzy set in U is a set of ordered pairs:

Ae^k_j ¼ fðx;l^k_A^jð ÞÞjxx 2Ug.

Example 4.4. According to Example 4.3, the r-semi fuzzy set of a subset A¼ fa;cg is

Ae^s_r ¼ ða;1Þ; b;1 3

;ðc;1Þ;ðd;0Þ

:

But ther-fuzzy set of a subsetA¼ fa;cg is Ae_r¼ ða;1Þ;b;¹₂

; c;²₃

;ðd;0Þ

.

5. Illustrative examples

The main goal of this section is to introduce two practical examples in order to illustrate the importance of applying near concept in rough context. In the first example we use an equivalence relation that induced from an information system and hence we compare between our approaches and Pawlak approach. In the sec- ond example, we apply our approaches in a multi-valued information system (MVIS)[14]. This type of information system is generalization to information sys- tem which uses an arbitrary binary relation and thus Pawlak approach does not fit in this type. Lin[10]introduced general rough membership function depending on an arbitrary binary relation, these rough membership function coincide with ourj- rough membership function in the case ofj¼ronly. But, the other typesjof ourj- rough membership functions are more accurate thanj¼r, so we can see that our approaches are the appropriate tools for these types and very useful in information analysis. Finally, in the second example we introduce a comparison between our approaches and Lin method.

Example 5.1. Consider the following information system as in Table 5.1 that represents the data about 6 students, as shown below.

FromTable 5.1, we have.

The set of universe: U¼ f1;2;3;4;5;6g,

The set of attributes: AT¼fAnalysis; Algebra; Statisticsg ¼ C[fDecisiong ¼D,

The sets of values: V_Analysis¼ fBad; Goodg;V_Algebra ¼ fBad; Goodg, V_Statistics¼ fBad; Medium; Goodg and V_Decision¼ fAccept; Rejectg.

But we take the set of condition attributes, C¼fAnalysis; Algebra;

Statisticsg.

Thus we have:U=C¼ff g;1 f2;5g;f g;3 f g;4 f g6 gand the set of r-pre rough set is

P_rð Þ ¼U }ð Þ ðset of all subsets in UÞ:U

Suppose that XðDecision:AcceptÞ ¼ f1;2;3;6g. Thus we compute the rough membership function with respect to Pawlak [13,14] and with respect to our approaches as follows:

Pawlak Definition[13,14] (rough membership function):

Forx¼1, thenl^C_Xð Þ ¼1 ^jf1g\X_jf1gj^j¼1. Forx¼3, thenl^C_Xð Þ ¼3 ^jf3g\X_jf3gj^j¼1.

Forx¼2, thenl^C_Xð Þ ¼2 ^jf2;5g\X_jf2;5gj^j¼¹₂. Forx¼6, thenl^C_Xð Þ ¼6 ^jf6g\X_jf6gj^j¼1.

Our Definition (r-pre rough membership function):

W^p_X^rð Þ ¼1 jf1g \Aj f1g

j j ¼1; jf1;2g \Aj f1;2g

j j ¼1;. . .

) l^p_X^rð Þ ¼1 1;

W^p_X^rð Þ ¼2 jf2g \Aj f2g

j j ¼1; jf2;6g \Aj f2;6g

j j ¼1;. . .

) l^p_X^rð Þ ¼2 1;

W^p_X^rð Þ ¼3 jf3g \Aj f3g

j j ¼1; jf3;5g \Aj f3;5g

j j ¼1

2;. . .

) l^p_X^rð Þ ¼3 1 and

Table 5.1 Information system.

W^p_X^rð Þ ¼6 jf6g \Aj f6g

j j ¼1; jf6;5g \Aj f6;5g

j j ¼1

2;. . .

) l^p_X^rð Þ ¼6 1:

Moreover, for some elements that has decision (Reject) such that 5 we get:

In Pawlak: l^C_Xð Þ ¼5 ^jf_j^2;5_f2;5^g\X_gj^j¼¹₂, that is 5 may be belongs to the set XðDecision:AcceptÞ,

X¼ f1;2;3;6g and this contradicts to Table 5.1.

But in our definition: we have

W^p_X^rð5Þ ¼n^jf5g\A_jf5gj^j¼0; ^jf5;6g\A_jf5;6gj^j¼¹₂;. . .o

) l^p_X^rð Þ ¼5 0.

This means that 5 does not belongs to the setXðDecision:AcceptÞ ¼ f1;2;3;6g which is coincide with Table 5.1. Hence, our approaches are more accurate than Pawlak definition.

Example 5.2. Consider the following multi-valued information system (MVIS) as inTable 5.2. Suppose we are given data about 5 persons, as shown below.

Where R₁¼ Languages¼ fEnglish; German; Arabicg, R₂¼ Sports¼ Handball; Basketball; Tennis

f g and R3¼ Skills¼

fSwimming; Running; Fishingg such that xRny; 8n¼1;2;3.

We will use the case ofj¼r and k¼c as follows:

aR1 ¼fa;bg; bR1¼f g;b cR1¼fb;c;dg; dR1¼f g;d eR1 ¼fd;eg;

aR₂ ¼fa;b;cg; bR₂ ¼fa;b;cg; cR₂ ¼f g;c dR₂ ¼fc;dg; eR₂¼f ge and aR3 ¼fa;bg; bR3¼fa;bg; cR3 ¼fc;dg; dR3¼f g;d eR3 ¼fd;eg:

In order to represent the set of all condition attributes, we generate the following relation from all above relations as follows: xR ¼T3

n¼1xR_n. Thus we get aR ¼fa;bg; bR ¼f g;b cR ¼fc;dg; dR ¼f g;d eR ¼f g.e

Clearly, this relation is symmetry relation (reflexive and symmetric) but is not transitive and thus it is not equivalence relation. Hence, Pawlak approach does not fit in this case, so we use Lin definition and our approaches as follows:

Table 5.2 Multi-valued information system (MVIS).

Suppose thatXðDecision:AcceptÞ ¼ fa;c;dg. Thus.

Lin Definition[10] (rough membership function):

Forx¼a, thenl^R_Xð Þ ¼b ^jf_j^a;b_fa;b^g\X_gj^j¼1.

Forx¼c, thenl^R_Xð Þ ¼c ^jf_j^c;d_fc;d^g\X_gj^j¼¹₂. Forx¼d, thenl^R_Xð Þ ¼d ^{jf g\X}_j^d_{f gdj}^j¼1.

Our approaches(G_n–CAS):

First, the relation R is R ¼ fða;aÞ;ða;bÞ;ðb;bÞ; ðc;cÞ;ðc;dÞ;ðd;dÞ;ðe;eÞg.

Thus we get.

The r-neighborhoods of all elements are: Nrð Þ ¼a fa;bg, Nrð Þ ¼b f g,b Nrð Þ ¼c fc;dg, Nrð Þ ¼d f g,d Nrð Þ ¼e f g.e

The l-neighborhoods of all elements are: Nlð Þ ¼a f g,a Nlð Þ ¼b fa;bg, Nlð Þ ¼c f g,c Nlð Þ ¼d fc;dg,Nlð Þ ¼e f g.e

Thei-neighborhoods of all elements are:Nið Þ ¼a f g,a Nið Þ ¼b f g,b Nið Þ ¼c f g,c Nið Þ ¼d f g,d Nið Þ ¼e f g.e

1.r-rough membership function:

Forx¼a, thenl^r_Xð Þ ¼a ^jfa;b_jfa;b^g\X_gj^j¼1.

Forx¼c, thenl^r_Xð Þ ¼c ^jf_j^c;d_fc;d^g\X_gj^j¼¹₂. Forx¼d, thenl^r_Xð Þ ¼d ^{jf g\Xd}_{jf gdj}^j¼1.

2.l-rough membership function:

Forx¼a, thenl^l_Xð Þ ¼a ^{jf g\Xa}_{jf gaj}^j¼1.

Forx¼c, thenl^l_Xð Þ ¼c ^{jf g\X}_j^c_{f gcj}^j¼1.

Forx¼d, thenl^l_Xð Þ ¼d ^jfc;d_jfc;d^g\X_gj^j¼1.

3.i-rough membership function:

Forx¼a, thenlⁱ_Xð Þ ¼a ^{jf g\Xa}_{jf gaj}^j¼1.

Forx¼c, thenlⁱ_Xð Þ ¼c ^{jf g\X}_j^c_{f gcj}^j¼1.

Forx¼d, thenlⁱ_Xð Þ ¼d ^{jf g\Xd}_{jf gdj}^j¼1.

It is clear that Lin rough membership function is the same asr-rough member- ship function. Moreover, our approaches l-rough (resp. i-rough) membership function is more accurate thanr-rough membership function and Lin rough mem- bership function. Finally, we can also applyj-near rough membership function as inExample 5.1.

6. Conclusions and future works

In this work, we introduced one of an important topological application that named ‘‘near concepts’’ in rough context. Accordingly, different types of approx- imations (resp. rough membership relations and functions) were provided to be easy mathematical tools to classify the sets and help for measuring exactness and roughness of sets. These tools are more accurate than other types that were defined by others authors. Consequently, our approaches are very interesting in decision making. We believe that these structures are useful in the applications and thus these techniques open the way for more topological applications in rough context and help in formalizing many applications from real-life data. In our future works, we will apply the suggested methods in this paper in real life appli- cations and problems.

Acknowledgements

The authors would like to thank the anonymous referees and the Editor-in-Chief, Professor Hatim Aboalsamh for their valuable suggestions in improving this paper.

References

[1]B. Chen, J. Li, On topological covering-based rough spaces, Int. J. Phys. Sci. 6 (17) (2011) 4195–4202.

[2]G. Liu, Y. Sai, A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason. 50 (2009) 521–528.

[3]J.A. Pomykala, Approximation operators in approximation space, Bull. Polish Acad. Sci. Math. 36 (1988) 193–210.

[4]Lijuan Wang, Xibei Yang, Chen Wu, Multi-covering based rough set model, rough sets, fuzzy sets, data mining, and granular computing, Lect. Notes Comput. Sci. 8170 (2013) 236–244.

[5]M.E. Abd El-Monsef, O.A. Embaby, M.K. El-Bably, New Approach to covering rough sets via relations, Int. J. Pure Appl. Math. 91 (3) (2014) 329–347.

[6] M.E. Abd El-Monsef, A.M. Kozae, M.K. El-Bably, On generalizing covering approximation space, J. Egypt.

Math. Soc. (2015),http://dxdoi.org/10.1016/j.joems.2014.12.007.

[7]M. Restrepo, J. Go´mez, Covering based rough sets and relation based rough sets, rough sets and intelligent systems paradigms, Lect. Notes Comput. Sci. 8537 (2014) 143–152.

[8]P. Zhu, Covering rough sets based on neighborhoods: an approach without using neighborhoods, Int. J.

Approx. Reason. 52 (3) (2011) 461–472.

[9]R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Trans. Knowl. Data Eng. 12 (2) (2000) 331–336.

[10]T.Y. Lin, Granular computing on binary relation I: Data mining and neighborhood systems, in: Rough Sets in Knowledge Discovery, Physica-Verlag, 1998, pp. 107–121.

[11]W. Zhu, Topological approaches to covering rough sets, Int. J. Comput. Inform. Sci. 177 (2007) 1499–1508.

[12]Y.Y. Yao, B. Yao, Covering based rough set approximations, Inform. Sci. 200 (2012) 91–107.

[13]Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982) 341–356.

[14]Z. Pawlak, Rough Sets, Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Boston, 1991.