Generalized covering approximation space and near

ACI 35 25 March 2015

No. of Pages 25

Applied Computing and Informatics (2015) xxx, xxx–xxx 1

Saudi Computer Society, King Saud University

Applied Computing and Informatics (http://computer.org.sa) www.ksu.edu.sa www.sciencedirect.com

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ORIGINAL ARTICLE

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Generalized covering approximation space and near concepts with some applications

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

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

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Received 27 May 2014; revised 10 February 2015; accepted 11 February 2015

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*

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KEYWORDS

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Coverings; Generalized covering approximation space; Near concepts; Memberships relations and functions; Fuzzy sets

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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 generalized covering approximation space. Accordingly, several types of fuzzy sets are constructed. The basic notions of near approximations are introduced and sufficiently 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. ª 2015 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: [email protected] (M.E. Abd El-Monsef), [email protected] (A.M. Kozae), mkamel_ [email protected] (M.K. El-Bably). Peer review under responsibility of King Saud University.

Production and hosting by Elsevier http://dx.doi.org/10.1016/j.aci.2015.02.001 2210-8327 ª 2015 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/). Please cite this article in press as: M.E. Abd El-Monsef et al., Generalizedcovering approximation space and near concepts with some applications, Applied Computing and Informatics (2015), http://dx.doi.org/10.1016/j.aci.2015.02.001


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M.E. Abd El-Monsef et al.

1. Introduction

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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 classical rough sets by relaxing the partitions arising from equivalence relations to coverings. 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 space Gn – CAS as a generalization to rough set theory and covering approximation space. 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 relations and functions and new types of fuzzy sets in Gn – CAS. Second part aims to introduce one of an important topological concepts which are called ‘‘near concepts’’ in rough context (specially, in Gn – CAS). In fact, we apply near concepts in Gn – CAS to define different tools for modifying the original operations. The suggested methods in this paper represent easy mathematical tools to approximate the rough sets and removing the uncertainty (vagueness) of some rough sets. In addition, comparisons 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 importance 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.

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2. Preliminaries

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In this section, we introduce the fundamental concepts that were used through this paper. Definition 2.1 [11]. Let U be a domain of discourse, S C ¼ fCk jk 2 Kg a family of subsets of U. If none subsets in C is empty, and k2K Ck ¼ U, then C is called a covering of U. The pair hU; Ci is called a ‘‘covering approximation space’’ if C is a covering of U. Definition 2.2 [11]. Let hU; Ci be a ‘‘covering approximation space’’. For any x 2 U, we define the neighborhood of x as follows: NðxÞ ¼ \fK 2 Cjx 2 Kg. Definition 2.3 [11]. Let hU; Ci be a covering approximation space. For any X 2 U, the lower approximation of X is defined as: X ¼ [fKjK 2 C and K # Xg.

Please cite this article in press as: M.E. Abd El-Monsef et al., Generalizedcovering approximation space and near concepts with some applications, Applied Computing and Informatics (2015), http://dx.doi.org/10.1016/j.aci.2015.02.001


Generalized covering approximation space and near concepts 60

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And the upper approximation of X is defined as: X* = X* [{N(x)|x 2 X X*}. Definition 2.4 ([13,14]). Let U be a non-empty set and B be an equivalence relation on U. Thus for every X ˝ U, Pawlak rough membership function is lBX : U ! ½0; 1 and lBX ðxÞ ¼ j½xj½xB \Xj where |X| denotes the cardinality of X and Bj [x]B represent the equivalence class of element x 2 U. The rough membership function lBX expresses conditional probability that x belongs to X given B and can be interpreted as a degree that x belongs to X in view of information about x expressed by B.

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Definition 2.5 [10]. Let U be a non-empty set and R be an equivalence relation on U. Thus for every X 2 U, Lin rough membership function is lR X : U ! ½0; 1 and jxR\Xj R lX ðxÞ ¼ jxRj where xR represent the after set of element x 2 U.

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3. Generalized covering approximation space

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In this section, we outline the main ideas about the generalized covering approximation space ‘‘Gn – covering approximation space’’, Gn – CAS, that was given in [6]. Many results, examples and counter examples are provided. Definition 3.1 [6]. Let U be any set, and R be any binary relation on U. Then the ‘‘after set’’ (resp. ‘‘fore set’’) of the element x 2 U is the class xR = {y 2 U:xRy} (resp. Rx = {y 2 U:yRx}). Definition 3.2. Let U „ B be a finite set and R be a binary relation on U. Then, we can define two different coverings for U induced from the binary relation R as follows: S (i) Right Covering (briefly, r-cover): Cr ¼ fxR : 8x 2 U g such that U ¼ S x2U xR. (ii) Left Covering (briefly, l-cover): Cl ¼ fRx : 8x 2 U g such that U ¼ x2U Rx. Definition 3.3. Let U „ B be a finite set, R be a binary relation on U and Cn be n-cover of U associated to R, where n 2 {r, l}. Then the triple hU; R; Cn i is called ‘‘ Gn – covering approximation space’’ (briefly, Gn – CAS). Definition 3.4. Suppose that the triple hU; R; Cn i be Gn – CAS. For every element x 2 U, we can define four different neighborhoods Nj(x), as follows: For each j 2 {r, l, i, u} (i) r-neighborhood: N r ðxÞ ¼ \fK 2 Cr jx 2 Kg. (ii) l-neighborhood: N l ðxÞ ¼ \fK 2 Cl jx 2 Kg.



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(iii) i-neighborhood: N i ðxÞ ¼ N r ðxÞ \ N l ðxÞ. (iv) u-neighborhood: N u ðxÞ ¼ N r ðxÞ [ N l ðxÞ.

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Definition 3.5. Suppose the triple hU; R; Cn i be Gn – CAS. For each j 2 {r, l, i, u} and A ˝ U, the j-lower and the j-upper approximations of A are defined respectively as follows:

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Rj ðAÞ ¼ fx 2 AjNj ðxÞ # Ag

and Rj ðAÞ ¼ fx 2 UjNj ðxÞ \ A – £g:

It is clear that Rj ðAÞ # Rj ðAÞ; 8A # U. Definition 3.6. Suppose the triple hU; R; Cn i be Gn – CAS and A ˝ U. Thus, for each j 2 {r, l, i, u}, the subset A is called ‘‘j-exact’’ set if Rj ðAÞ ¼ Rj ðAÞ ¼ A. Otherwise, A is called ‘‘j-rough’’ set. Definition 3.7. Suppose the triple hU; R; Cn i be Gn – CAS. For each j 2 {r, l, i, u} and A ˝ U, the j-boundary, j-positive and j-negative regions of A are defined respectively as follows:

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Bj ðAÞ ¼ Rj ðAÞ Rj ðAÞ;

POSj ðAÞ ¼ Rj ðAÞ

and NEGj ðAÞ ¼ U Rj ðAÞ:

Definition 3.8. Suppose the triple hU; R; Cn i be Gn – CAS. For each j 2 {r, l, i, u} and A ˝ U, the j-accuracy of the approximations of A is defined as follows:

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dj ðAÞ ¼ 119

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jRj ðAÞj ; jRj ðAÞj

where Rj ðAÞ – 0 and jAj denotes the cardinality of A:

Remark 3.1. From the above definitions, we notice that:

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(i) Obviously, 0 6 dj ðAÞ 6 1, for every A ˝ U. (ii) A is j-exact set if dj(A) = 1 and Bj ðAÞ ¼ £. Otherwise, it is j-rough set.

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

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Proposition 3.1. Let the triple hU; R; Cn i be; and A ˝ U. Then

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(i) (ii) (iii) (iv)

Ru ðAÞ # Rr ðAÞ # Ri ðAÞ. Ru ðAÞ # Rl ðAÞ # Ri ðAÞ. Ri ðAÞ # Rr ðAÞ # Ru ðAÞ. Ri ðAÞ # Rr ðAÞ # Ru ðAÞ.



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Generalized covering approximation space and near concepts

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Corollary 3.1. Let the triple hU; R; Cn i be Gn – CAS and A ˝ U. Then (i) Bi ðAÞ # Br ðAÞ # Bu ðAÞ. (ii) Bi ðAÞ # Bl ðAÞ # Bu ðAÞ.

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Corollary 3.2. Let the triple hU; R; Cn i be Gn – CAS and A ˝ U. Then (i) du ðAÞ 6 dr ðAÞ 6 di ðAÞ. (ii) du ðAÞ 6 dl ðAÞ 6 di ðAÞ. Proposition 3.2. Suppose that the triple hU; R; Cn i be Gn – CAS and A ˝ U. Then the following statements are true in general: (i) A is u-exact ) A is r-exact ) A is i-exact. (ii) A is u-exact ) A is l-exact ) A is i-exact.

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The proof of the above results can found in [6]. 4. j-Rough membership relations, j-rough membership functions and j-fuzzy sets

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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 in Gn – CAS. Moreover, the suggested rough membership relations (resp. functions) are more accurate than classical rough membership function that was given by Lin [10] and the other types.

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Definition 4.1. Let hU; R; Cn i be Gn – CAS and A ˝ U. Then we say that:

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(i) x is ‘‘j-surely’’ belongs to A, written x 2j A, if x 2 Rj ðAÞ. (ii) x is ‘‘J-possibly’’ belongs to X, written x 2j A, if x 2 Rj ðAÞ.

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These two rough membership relations are called ‘‘j-strong’’ and ‘‘j-weak’’ membership relations respectively. Lemma 4.1. Suppose that the triple hU; R; Cn i be Gn – CAS and A ˝ U. Then the following statements are true in general: (i) x 2j A implies x 2 A. (ii) x 2 A implies x 2j A.



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Proof. Straightforward.

h

The converse of the above lemma is not true in general, as the following example illustrates: Example 4.1. Consider the triple hU; R; Cn i be Gn – CAS, where U = {a, b, c, d} and R ¼ fða; aÞ; ðb; bÞ; ðc; cÞ; ðc; bÞ; ðc; dÞ; ðd; aÞg. We will show the above remark in case of j = r and the other cases similarly. Suppose that A = {a, b, d}, then we get aR ¼ fag; bR ¼ fbg; cR ¼ fb; c; dg ¼ dR and Nr(a) = {a}, Nr(b) = {b}, Nr(c) = {b, c, d} = Nr(d). Thus Rr ðAÞ ¼ fa; bg and Rr ðAÞ ¼ U. Clearly d 2 A but d Rr A and c 2r A but c R A. The following proposition is very interesting since it 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 4.1. Let hU; R; Cn i be Gn – CAS and A ˝ U. Then the following are true generally (i) (ii) (iii) (iv)

If If If If

x 2i A ) x 2r A ) x 2u A. x 2i A ) x 2l A ) x 2u A. x 2u A ) x 2r A ) x 2i A. x 2u A ) x 2l A ) x 2i A.

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Proof. We will prove the first statement and the others similarly: (i) If x 2i A ) x 2 Ri ðAÞ ) x 2 Rr ðAÞ ) x 2r A:

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Also, if x 2r A ) x 2 Rr ðAÞ ) x 2 Ru ðAÞ ) x 2u A. h The converse of the above proposition is not true in general as the following example illustrates. Example 4.2. Let the triple hU; R; Cn i be Gn – CAS, where U = {a, b, c, d} and R ¼ fða; aÞ; ða; bÞ; ðb; cÞ; ðb; dÞ; ðc; aÞ; ðd; aÞg. Then we get Nr(a) = {a}, Nr(b) = {a, b}, Nr(c) = {c, d} = Nr(d). Nl(a) = {a}, Nl(b) = {b}, Nl(c) = {a, c, d} = Nr(d). Nu(a) = {a}, Nu(b) = {a, b}, Nu(c) = {a, c, d} = Nu(d). Ni(a) = {a}, Ni(b) = {b}, Ni(c) = {c, d} = Ni(d). Suppose that A = {b, c, d}. Then Ru ðAÞ ¼ £, Rr ðAÞ ¼ fc; dg, Rl ðAÞ ¼ fbg and Ri ðAÞ ¼ fb; c; dg. Accordingly, c 2r A and b 2l A but b Ru A and c Ru A. Also b 2i A and c 2i A but b Rr A and c Rl A.



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By similar way, we can illustrate the others cases.

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Definition 4.2. Let hU; R; Cn i be Gn – CAS and A ˝ U. Then for each j 2 {r, l, i, u} and x 2 U:

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The j-rough membership functions on U for subset A are lAj : U ! ½0; 1 where

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lAj ðxÞ ¼

jNj ðxÞ\Aj jNj ðxÞj

and |A| denotes the cardinality of A.

The rough j-membership function expresses conditional probability that x belongs to A given R and can be interpreted as a degree that R belongs to A in view of information about x expressed by R. Moreover, in case of infinite universe, the above membership function lAj can be use for spaces having locally finite minimal neighborhoods for each point. Remark 4.1. The rough j-membership functions can be used to define japproximations of a set A, as follows:

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Rj ðAÞ ¼ fx 2 UjlAj ðxÞ ¼ 1g

and Rj ðAÞ ¼ fx 2 UjlAj ðxÞ > 0g:

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

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Proposition 4.2. Let hU; R; Cn i be Gn – CAS and A, B ˝ U. Then

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(i) (ii) (iii) (iv) (v) (vi)

lAj ðxÞ ¼ 1 () x 2j A. lAj ðxÞ ¼ 0 () x 2 U Rj ðAÞ. j ðxÞ P max lAj ðxÞ; lBj ðxÞ for any x 2 U . lA[B 0 < lAj ðxÞ < 1 () x 2 Bj ðAÞ. lUj A ðxÞ ¼ 1 lAj ðxÞ for any x 2 U . j ðxÞ 6 min lAj ðxÞ; lBj ðxÞ for any x 2 U. lA\B

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

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x2j A () x 2 Rj ðAÞ () x 2 A; Nj ðxÞ # A () lAj ðxÞ ¼ 1:

Remark 4.2. The rough j-membership functions divide the universe U by using the j-boundary, j-positive and j-negative regions of A ˝ U, respectively as follows:

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Bj ðAÞ ¼ fx 2 Uj0 < lAj ðxÞ < 1g; 243

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¼ 1g

POSj ðAÞ ¼ fx 2 UjlAj ðxÞ

and NEGj ðAÞ ¼ fx 2 UjlAj ðxÞ ¼ 0g:

Lemma 4.2. Let hU; R; Cn i be Gn – CAS and A ˝ U. Then for every x 2 U



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(i) luA ðxÞ ¼ 1 ) lrA ðxÞ ¼ 1 ) liA ðxÞ ¼ 1: (ii) luA ðxÞ ¼ 1 ) llA ðxÞ ¼ 1 ) liA ðxÞ ¼ 1:

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Proof. (i) luA ðxÞ ¼ 1 ) x 2u A ) x 2r A ) lrA ðxÞ ¼ 1. Also, lrA ðxÞ ¼ 1 ) x 2r A ) x 2i A ) liA ðxÞ ¼ 1. (ii) Similarly as (i). h Lemma 4.3. Let hU; R; Cn i be Gn – CAS and A ˝ U. Then for every x 2 U (i) luA ðxÞ ¼ 0 ) lrA ðxÞ ¼ 0 ) liA ðxÞ ¼ 0. (ii) luA ðxÞ ¼ 0 ) llA ðxÞ ¼ 0 ) liA ðxÞ ¼ 0.

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Proof. (i) luA ðxÞ ¼ 1 ) Nu ðxÞ \ A ¼ £ ) Nr ðxÞ \ A ¼ £ ) lrA ðxÞ ¼ 0.

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Also, lrA ðxÞ ¼ 0 ) Nr ðxÞ \ A ¼ £ ) Ni ðxÞ \ A ¼ £ ) liA ðxÞ ¼ 0.

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(ii) Similarly as (i). h Remark 4.3. According to the above results and by using Proposition 4.2, we can prove that liA is more accurate than the others types, this means that: (i) If x 2 A ) luA ðxÞ 6 lrA ðxÞ 6 liA ðxÞ and if x 2 A ) luA ðxÞ 6 llA ðxÞ 6 liA ðxÞ. (ii) If x R A ) liA ðxÞ 6 lrA ðxÞ 6 luA ðxÞ and if x 2 A ) liA ðxÞ 6 llA ðxÞ 6 luA ðxÞ.

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The converse of the above lemmas is not true in general.

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The following example illustrates Remark 4.4.

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Example 4.3. According to Example 4.2, consider the subset A ¼ fb; c; dg. Then we get

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lrA ðaÞ ¼ jfag\Aj ¼ 0: lrA ðbÞ ¼ jfa;bg\Aj ¼ 12 : lrA ðcÞ ¼ jfc;dg\Aj ¼ 1: jfagj jfa;bgj jfc;dgj ¼ 0: llA ðbÞ ¼ jfbg\Aj ¼ 1: llA ðaÞ ¼ jfag\Aj jfagj jfbgj

lrA ðdÞ ¼ jfc;dg\Aj ¼ 1: jfc;dgj

llA ðcÞ ¼ jfa;c;dg\Aj ¼ 23 : llA ðdÞ ¼ jfa;c;dg\Aj ¼ 23 : jfa;c;dgj jfa;c;dgj

¼ 0: luA ðbÞ ¼ jfa;bg\Aj ¼ 12 : luA ðcÞ ¼ jfa;c;dg\Aj ¼ 23 : luA ðdÞ ¼ jfa;c;dg\Aj ¼ 23 : luA ðaÞ ¼ jfag\Aj jfagj jfa;bgj jfa;c;dgj jfa;c;dgj 273 274

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liA ðaÞ ¼ jfag\Aj ¼ 0: liA ðbÞ ¼ jfbg\Aj ¼ 1: jfagj jfbgj

liA ðcÞ ¼ jfc;dg\Aj ¼ 1: jfc;dgj

liA ðdÞ ¼ jfc;dg\Aj ¼ 1: jfc;dgj

Remark 4.4. Lin [10] have defined rough membership function for any binary relation, this membership function coincide with our membership function lAj , in case



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of j ¼ r (r-rough membership function lrA ) only. Therefore, our approaches represent generalization for Lin approaches. Moreover, our membership functions are more accurate than Lin membership function. 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 lAj to define four different types of fuzzy sets in Gn – CAS. Definition 4.3. Let hU; R; Cn i be Gn – CAS and A # U. Then the j-fuzzy sets in U is the set of ordered pairs:

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A~j ¼ fðx; lAj ðxÞÞjx 2 Ug;

8j 2 fr; l; i; ug:

Example 4.4. According to Example 4.3, consider the subset A ¼ fb; c; dg. Then we get 1 2 2 ~ ~ ; Ar ¼ ða; 0Þ; b; ; ðc; 1Þ; ðd; 1Þ ; Al ¼ ða; 0Þ; ðb; 1Þ; c; ; d; 2 3 3 1 2 2 ~ Au ¼ ða; 0Þ; b; ; c; ; d; ; and A~i ¼ fða; 0Þ; ðb; 1Þ; ðc; 1Þ; ðd; 1Þg: 2 3 3 5. Near rough concepts in the generalized covering approximation space Gn CAS The main goal of this section is to introduce one of the important topological applications which are named ‘‘near concepts’’ in Gn – CAS. Moreover, we introduce 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 and j-accuracy measures). In addition, we introduce near exactness and near roughness by applying near concepts to make more accuracy for definability of sets in Gn – CAS. Definition 5.1. Suppose that hU; R; Cn i be Gn – CAS and A # U. Thus we define ‘‘near rough’’ sets in U as follows: For each j 2 fr; l; i; ug, the subset A is called (i) j-Pre rough set (briefly pj ) if A # Rj Rj ðAÞ . (ii) j-Semi rough set (briefly sj ) if A # Rj Rj ðAÞ . (iii) cj -rough set if A # Rj Rj ðAÞ [ Rj Rj ðAÞ .

The above sets are called ‘‘j-near rough sets’’ and the families of j-near rough sets of U denotes by Kj(U), for each K 2 fP; S; cg.



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Remark 5.1. The family of j-pre rough sets and the family of j-semi rough sets are not comparable as the following example illustrates.

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Example 5.1. Let the triple hU; R; Cn i be Gn – CAS, where U ¼ fa; b; c; dg and

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R ¼ fða; 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 j = r and the other cases similarly as follows:

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Nr ðaÞ ¼ fa; bg ¼ Nr ðbÞ;

Nr ðcÞ ¼ U;

Nr ðdÞ ¼ fdg:

Thus we compute the j-near rough sets for j = r as follows: The family of r-pre rough sets is: Pr ðUÞ ¼ fU; £; fag; fbg; fdg; fa; bg; fa; dg; fb; dg; fa; b; dg; fa; c; dg; fb; c; dgg. The family of r-semi rough sets is: Sr ðUÞ ¼ fU; £; fdg; fa; bg; fc; dg; fa; b; cg; fa; b; dgg. The main goal of the following definitions is to introduce the new approximation operators (j-near approximations) which modify and generalize the japproximations. Definition 5.2. Let hU; R; Cn i be Gn – CAS and A # U. Then we define the j-near approximations of any subset A as follows: For each j 2 fr; l; i; ug (i) The j-pre lower and the j-pre upper approximations of A are defined respectively by Rpj ðAÞ ¼ fx 2 AjNj ðxÞ # Rj ðAÞg and Rpj ðAÞ ¼ A [ fx

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2 Ac jNj ðxÞ \ Rj ðAÞ – £g: (ii) The j-semi lower and the j-semi upper approximations of A are defined respectively by

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Rsj ðAÞ ¼ fx 2 AjNj ðxÞ \ Rj ðAÞ – £g 349 350 351 352

and Rsj ðAÞ ¼ A [ fx

2 Rj ðAÞjNj ðxÞ # Rj ðAÞg: (iii) The cj -lower and the cj -upper approximations of A are defined respectively by

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Rcj ðAÞ ¼ Rpj ðAÞ [ Rsj ðAÞ and Rcj ðAÞ ¼ Rpj ðAÞ \ Rsj ðAÞ: The following proposition introduces the fundamental properties of the j-near approximations.



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Proposition 5.1. Let hU; R; Cn i 8j 2 fr; l; i; ug; k 2 fp; s; cg:

be

Gn – CAS

(10) Rkj Rkj ðAÞ ¼ Rkj ðAÞ. (11) Rkj Rkj ðAÞ ¼ Rkj ðAÞ. Proof. Firstly, the proof of (1), (2), (10) and (11) is obvious directly from Definition 5.2. Now, we will prove the left properties for case k ¼ p and the other cases similarly. (3) If A # B Rj ðBÞg ¼ Rpj ðBÞ.

then

Rpj ðAÞ ¼ fx 2 AjNj ðxÞ # Rj ðAÞg # fx 2 BjNj ðxÞ #

(9) From Lemma 5.1, we get

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Then,

of A:

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386

A; B # U.

Rkj ðAÞ # A # Rkj ðAÞ. Rkj ðU Þ ¼ Rkj ðU Þ ¼ U and Rkj ð£Þ ¼ Rkj ð£Þ ¼ £. If A # B then Rkj ðAÞ # Rkj ðBÞ. If A # B then Rkj ðAÞ # Rkj ðBÞ. Rkj ðA \ BÞ # Rkj ðAÞ \ Rkj ðBÞ. Rkj ðA \ BÞ # Rkj ðAÞ \ Rkj ðBÞ. Rkj ðA [ BÞ Rkj ðAÞ [ Rkj ðBÞ. Rkj ðA [ BÞ Rkj ðAÞ [ Rkj ðBÞ. h ic h ic (9) Rkj ðAÞ ¼ Rkj ðAc Þ and Rkj ðAÞ ¼ Rkj ðAc Þ ; where Ac is the complement

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

385

and

(1) (2) (3) (4) (5) (6) (7) (8)

381

383

11

p c c c c Rj ðA Þ ¼ A \ Rj Rj ðAÞ ¼ Ac [ Rj Rj ðAÞ ¼ Ac [ Rj Rj ðAÞ ¼ Rpj ðAÞ: c Similarly, Rpj ðAÞ ¼ Rpj ðAc Þ .

h

Definition 5.3. Let hU; R; Cn i be Gn – CAS and A # U. Then we define the j-near boundary, j-near positive and j-near negative regions of A are defined respectively as follows:

390

8j 2 fr; l; i; ug; k 2 fp; s; cg : Bkj ðAÞ ¼ Rkj ðAÞ Rkj ðAÞ; 392

POSkj ðAÞ

¼ Rkj ðAÞ and NEGkj ðAÞ ¼ U Rkj ðAÞ:



No. of Pages 25

12 393 394 395 396

398

399 400 401 402 403 404

405 406 407 409 408

410 411 412 413 414

415

416 417 418 419 420

421 422


Definition 5.4. Let hU; R; Cn i be Gn – CAS and A # U. Then, for each j 2 fr; l; i; ug; k 2 fp; s; cg, the j-near accuracy of the j-near approximations of A # U is defined by k Rj ðAÞ k ; where Rkj ðAÞ – 0: Obviously 0 dkj ðAÞ 1: dj ðAÞ ¼ k Rj ðAÞ

Definition 5.5. Let hU; R; Cn i be Gn – CAS and A # U. Then, for each j 2 fr; l; i; ug; k 2 fp; s; cg, the subset A is called ‘‘j-near definable (briefly kj-exact) set’’ if Rkj ðAÞ ¼ Rkj ðAÞ ¼ A. Otherwise, it is called j-near rough (briefly kj-rough). It is clear that A is kj-exact if dkj ðAÞ ¼ 1 and Bkj ðAÞ ¼ £. Otherwise, it is kj-rough. Remark 5.2. In the Gn – CAS hU; R; Cn i, we can compute the j-near approximations of any subset A # U, directly by using the j-approximations, as the following lemma illustrates. Lemma 5.1. Let hU; R; Cn i be Gn – CAS and A # U. Then, for each j 2 fr; l; i; ug: (i) Rpj ðAÞ ¼ A \ Rj Rj ðAÞ . (ii) Rpj ðAÞ ¼ A [ Rj Rj ðAÞ . (iii) Rsj ðAÞ ¼ A \ Rj Rj ðAÞ . (iv) Rsj ðAÞ ¼ A [ Rj Rj ðAÞ . Proof. From Definition 5.2, the proof is obvious. h Lemma 5.2. Let hU; R; Cn i be Gn – CAS and A # U. Then, j 2 fr; l; i; ug; k 2 fp; s; cg: The subset A is kj-rough set if A ¼ Rkj ðAÞ.

for

each

The following results introduce the relationships between the j-approximations and the j-near approximations. Moreover, these results show the importance of applying near concepts in Gn – CAS. Theorem 5.1. Let hU; R; Cn i 8j 2 fr; l; i; ug; k 2 fp; s; cg:

be

Gn – CAS

and

A # U.

Then,

423 425

426 427

Rj ðAÞ # Rkj ðAÞ # A # Rkj ðAÞ # Rj ðAÞ Proof. We will prove the proposition in case of k = p and the other cases similarly.



No. of Pages 25

Generalized covering approximation space and near concepts 428 429

430 431 432

13

Let x 2 Rj ðAÞ, then x 2 A such that Nj ðxÞ # A. Thus x 2 A such that Nj ðxÞ # Rj ðAÞ and this implies x 2 Rpj ðAÞ. By duality, we get Rpj ðAÞ # Rj ðAÞ. h Corollary 5.1. Let hU; R; Cn i be Gn – CAS and A # U. Then, 8j 2 fr; l; i; ug; k 2 fp; s; cg:

433

(1) Bkj ðAÞ # Bj ðAÞ.

434

(2) dj ðAÞ 6 dkj ðAÞ.

435

436 437

438 439 440 441

Remark 5.3. 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 vagueness of sets and accordingly, these approaches is very useful in decision making.

442

443 444 445 447 448 449 450 451 452 453 454 455 456 457 458 459 460

461 462 463 464 465 466

Example 5.2. Let the triple hU; R; Cn i be Gn – CAS, where U ¼ fa; b; c; dg and R ¼ fða; aÞ; ða; bÞ; ðb; aÞ; ðb; bÞ; ðc; aÞ; ðc; bÞ; ðc; cÞ; ðc; dÞ; ðd; dÞg. Thus we get Nr ðaÞ ¼ fa; bg ¼ Nr ðbÞ;

Nr ðcÞ ¼ U;

Nr ðdÞ ¼ fdg:

By using Definitions 5.2 and 5.4, the following tables give comparisons between the j-approximations (resp. the j-accuracy of approximations) and the kj-approximations (resp. the kj-accuracy of approximations) of the all subsets of U, where j ¼ r; 8k 2 fp; s; cg: From Table 5.1, we notice that: There are four different methods to approximate the sets. The best of these methods is that given by using cr in constructing the approximations of sets, because the boundary regions decreased or cancelled by increasing the lower approximation and decreasing the upper approximation, that is Bcr ðAÞ # Bj ðAÞ. Accordingly, this will plays an important role for removing the vagueness (uncertainty) of rough sets. For example, the shaded sets in above tables are exact in cr but it is rough in the others types. From Table 5.2, we notice that: (i) Using cr in constructing the approximations of sets is more accurate than others types, since for any subset A # U , dr ðAÞ 6 dcr ðAÞ and dkr ðAÞ dcr ðAÞ; 8k 2 fp; sg. Thus, these approaches will helps to extract and discovery the hidden information in data that collected from real-life applications. For example, all shaded sets in the above table are exact sets in near approaches but it is rough in the approach dr.



14

No. of Pages 25


Table 5.1

Table 5.2



No. of Pages 25

Generalized covering approximation space and near concepts 469 467 468 470

471 472 473

474 475 476

477 478 479 480 481 482 483 484

485 486 487

488 489 490 491 493 494

15

(ii) Every r-exact set is r-near exact, but the converse is not true. For example, shaded sets in Tables 5.1 and 5.2. Remark 5.4. The following result is very interesting because it is prove that the jnear approaches are more accurate than the J-approaches. Moreover, it is illustrates the importance of j-near concepts in exactness of sets. Proposition 5.2. Let hU; R; Cn i be Gn – CAS and A # U. Then, 8j 2 fr; l; i; ug; k 2 fp; s; cg, the following is true in general: If A is j-exact set, then it is kj-exact. Proof. If A is j-exact set, then Bj ðAÞ ¼ £. Thus, by Corollary 5.1, Bkj ðAÞ ¼ £ and accordingly A is kj-exact. h The converse of the above proposition is not true in general as Example 5.3 illustrates. The main goal of the following results is to introduce the relationships between different types of j-near approximations, j-near boundary, j-near accuracy and jnear exactness respectively. Proposition 5.3. Let hU; R; Cn i be Gn – CAS and A # U. Then, 8j 2 fr; l; i; ug, the following statements are true in general: (i) (ii) (iii) (iv)

Rpj ðAÞ # Rcj ðAÞ. Rsj ðAÞ # Rcj ðAÞ. Rcj ðAÞ # Rpj ðAÞ. Rcj ðAÞ # Rsj ðAÞ.

492 495

496 497 498

499 500 501 503 502 504

505 506 507

Proof. By using Lemma 5.1 and 5.2, the proof is obvious.

h

Corollary 5.2. Let hU; R; Cn i be Gn – CAS and A # U. Then, 8j 2 fr; l; i; ug, the following statements are true in general: (i) (ii) (iii) (iv)

dpj ðAÞ 6 dcj ðAÞ. dSj ðAÞ 6 dcj ðAÞ. Bcj ðAÞ # Bpj ðAÞ. Bcj ðAÞ # Bsj ðAÞ.

Corollary 5.3. Let hU; R; Cj i be Gn – CAS and A # U. Then, 8j 2 fr; l; i; ug, the following statements are true in general:



16 508 511 509

No. of Pages 25


(i) A is pj -exact A is cj -exact. (ii) A is sj -exact A is cj -exact.

510 512

513 514 515

516 517

518 519 520 521 522 523

524 525 526

527 528 529 530

Remark 5.5. (i) The converse of the above results is not true in general as the following example illustrates. (ii) 8j 2 fr; l; i; ug; dcj ðAÞ ¼ maxðdpj ðAÞ; dSj ðAÞÞ, where max represents the maximum of two quantities. Example 5.3. According to Example 5.2, it is clear that A is cr-exact but it is not srexact. Also the subset B is cr-exact but it is not pr-exact. The relationships between different types of j-near approximations (for different types of j) are not comparable (no it is not like to j-approximations as Proposition 3.3) as the following remark illustrates. Remark 5.6. Let hU; R; Cj i be Gn – CAS and A # U. Then 8k 2 fp; s; cg, the following statements are not true in general: (i) (ii) (iii) (iv)

Rku ðAÞ # Rkr ðAÞ # Rki ðAÞ. Rku ðAÞ # Rkl ðAÞ # Rki ðAÞ. Rki ðAÞ # Rkr ðAÞ # Rku ðAÞ. Rki ðAÞ # Rkl ðAÞ # Rku ðAÞ.

531 532

533

The following example illustrates this remark. Example 5.4. Let the triple hU; R; Cj i be Gn – CAS, where U ¼ fa; b; c; dg and

534 536 537

R ¼ fða; aÞ; ðb; bÞ; ðb; aÞ; ðc; aÞ; ðc; dÞ; ðd; aÞ; ðd; cÞ; ðd; aÞg: Thus we get

538

Nr ðaÞ ¼ fag; Nr ðbÞ ¼ fa; bg; Nr ðcÞ ¼ fa; c; dg; Nr ðdÞ ¼ fdg; Nl ðaÞ ¼ U; Nl ðbÞ ¼ fbg; Nl ðcÞ ¼ fc; dg; Nl ðdÞ ¼ fdg; Nu ðaÞ ¼ fag; Nu ðbÞ ¼ fbg; Nu ðcÞ ¼ fc; dg; Nu ðdÞ ¼ fdg and 540 541 542 543 544

Ni ðaÞ ¼ U; Ni ðbÞ ¼ fa; bg; Ni ðcÞ ¼ fa; c; dg; Ni ðdÞ ¼ fdg: Now suppose that A ¼ fbg; B ¼ fa; dg and C ¼ fa; cg. Thus we get Rpr ðAÞ ¼ £, but Rpu ðAÞ ¼ A and Rpl ðBÞ ¼ fdg, but Rpu ðBÞ ¼ fa; dg. Also, we have Rsi ðCÞ ¼ fag, but Rsr ðCÞ ¼ A.



No. of Pages 25

Generalized covering approximation space and near concepts 545 546

547 548 549 550 551 552 553

554 555 556

557 558 559 560

17

6. j-near rough membership relations, j-near rough membership functions and j-near fuzzy sets in Gn CAS By considering j-near concepts, the new concepts ‘‘j-near rough membership relations’’ (resp. ‘‘j-near rough membership functions’’) are provided to modify and generalize the j-membership relations (resp. j-membership functions) in Gn – 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 6.1. Let hU; R; Cn i 8j 2 fr; l; i; ug; k 2 fp; s; cg, we say:

be

Gn – CAS

and

A # U.

Then

(i) x is ‘‘j-near surely’’ (briefly kj-surely) belongs to A, written x 2kj A, if x 2 Rkj ðAÞ. (ii) x is ‘‘j-near possibly’’ (briefly kj-possibly) belongs to X, written x 2j k A, if x 2 Rkj ðAÞ.

561

562 563 564

565 566

Lemma 6.1. Suppose the triple hU; R; Cn i be Gn – CAS and A # U. Thus 8j 2 fr; l; i; ug; k 2 fp; s; cg, the following statements are true in general: (i) If x 2kj A implies to x 2 A. (ii) If x 2 A implies to x 2kj A.

567

568 569 570

571 572

Proof. Straightforward. h The converse of the above lemma is not true in general, as the following example illustrates: Example 6.1. Consider Example 5.4, where U ¼ fa; b; c; dg and R ¼ fða; aÞ; ðb; bÞ; ðb; aÞ; ðc; aÞ; ðc; dÞ; ðd; aÞ; ðd; cÞ; ðd; aÞg. Thus we get

573 575 576 577 578

579 580

Nr ðaÞ ¼ fag;

Nr ðbÞ ¼ fa; bg;

Nr ðcÞ ¼ fa; c; dg;

Nr ðdÞ ¼ fdg:

We will show the above remark in case of ðj ¼ randk ¼ pÞ and the other cases similarly. Suppose that A ¼ fb; dg, then we get Rpr ðAÞ ¼ fdg and Rpr ðAÞ ¼ fb; c; dg. Clearly d 2 A but d Rpr A and c 2pr A but c R A.



No. of Pages 25

18 581 582

Remark 6.1. We can redefine the j-near approximations by using kj and 2kj as follows: For any A; B # U : Rkj ðAÞ ¼ fx 2 Ujxkj Ag and Rkj ðAÞ ¼ fx 2 Ujx2kj Ag.

583 584 585 586 587

588 589 590

591 593 592 594

595

The following proposition is very interesting since it is give the relationships between the j-rough membership relations and j-near rough membership relations. Accordingly, we will show the importance of using these different types of j-near rough membership relations. Proposition 6.1. Let hU; R; Cn i be Gn – CAS and A # U. 8j 2 fr; l; i; ug; k 2 fp; s; cg, the following statements are true in general:

598

599 600

Proof. We will prove first statement and the other similarly: (i) x 2j A ) x 2 Rj ðAÞ ) x 2 Rkj ðAÞ ) x 2kj A. h The converse of the above proposition is not true in general as the following example illustrates. Example 6.2. Let U ¼ fa; b; c; dg ðc; dÞ; ðd; aÞ; ðd; cÞ; ðd; aÞg.

603 604 605 606 607

608 609 610 611 612 615 614 616

618

and

R ¼ fða; aÞ; ðb; bÞ; ðb; aÞ; ðc; aÞ;

Thus we get Nr ðaÞ ¼ fag; Nr ðbÞ ¼ fa; bg; Nr ðcÞ ¼ fa; c; dg; Nr ðdÞ ¼ fdg.

601 602

Then

(i) x 2j A ) x 2kj A. (ii) x 2kj A ) x 2j A.

596 597


We will show the above remark in case of ðj ¼ r and k ¼ sÞ and the other cases similarly. Suppose that A ¼ fa; cg and B ¼ fb; dg, then we get Rr ðAÞ ¼ fag and Rsr ðAÞ ¼ fa; cg. Clearly c 2sr A; but c Rr A although c 2 A. Also Rr ðBÞ ¼ fb; c; dg and Rsr ðBÞ ¼ fb; dg. Clearly c Rsr B; but c 2r B although c R B. Definition 6.2. Suppose hU; R; Cn i be Gn – CAS and A # U. Thus we can define the j-near rough membership functions for Gn – CAS as follows: For each j 2 fr; l; i; ug; k 2 fp; s; cg and x 2 U, the j-near rough membership functions on U for subset A are k

lAj : U ! ½0; 1; where ( k 1 if 1 2 WAj ðxÞ: kj lA ðxÞ ¼ k minðWAj ðxÞÞ Otherwise:



No. of Pages 25

Generalized covering approximation space and near concepts k

619

620 621

and WAj ðxÞ ¼

jkj ðxÞ\Aj jkj ðxÞj

19

such that kj ðxÞ is a j-near rough set in that contains x.

Remark 6.2. The j-near rough membership functions can be used to define j-near approximations as follow:

622

k

624

Rkj ðAÞ ¼ fx 2 UjlAj ðxÞ ¼ 1g

k

and Rkj ðAÞ ¼ fx 2 UjlAj ðxÞ > 0g:

627

Lemma 6.2. Let hU; R; Cn i be Gn – CAS and A; B # U. Then, for each and j 2 fr; l; i; ug; k 2 fp; s; cg and x 2 U:Rj ðNj ðxÞÞ ¼ Rj ðNj ðxÞÞ ¼ Nj ðxÞ Rkj ðNj ðxÞÞ ¼ Rkj ðNj ðxÞÞ ¼ Nj ðxÞ.

628

Proof. First, from Lemma 2.3, if y 2 Nj ðxÞ ) Nj ðyÞ # Nj ðxÞ. Thus, we get

625 626

629 631 632

633 634

635 636 637 638 639

640 641 642

Rj ðNj ðxÞÞ ¼ fy 2 Nj ðxÞjNj ðyÞ # Nj ðxÞg ¼ Nj ðxÞ: Similarly Rj ðNj ðxÞÞ ¼ Nj ðxÞ. Also, by using Lemma k Rj ðNj ðxÞÞ ¼ Nj ðxÞ. h

5.1,

we

can

prove

that

Rkj ðNj ðxÞÞ ¼

Corollary 6.1. Let hU; R; Cn i be Gn – CAS and A; B # U. Then, for each j 2 fr; l; i; ug, k 2 fp; s; cg and x 2 U:Nj ðxÞ is j-near rough set. The following results give the fundamental properties of the j-near rough membership functions. Proposition 6.2. Let hU; R; Cn i be Gn – CAS and A; B # U. Then, for each j 2 fr; l; i; ug, k 2 fp; s; cg and x 2 U: k

643

(i) lAj ðxÞ ¼ 1 () x 2kj A. k

644

(ii) lAj ðxÞ ¼ 0 () x 2 U Rkj ðAÞ. k

645

(iii) 0 < lAj ðxÞ < 1 () x 2 Bkj ðAÞ. k

646 647 648

k

(iv) lUjA ðxÞ ¼ 1 lAj ðxÞ for any x 2 U . k k kj ðxÞ P maxðlAj ðxÞ; lBj ðxÞÞ for any x 2 U . (v) lA[B kj kj kj (vi) lA\B ðxÞ 6 minðlA ðxÞ; lB ðxÞÞ for any x 2 U.

649

650

651 652

Proof. We will prove (i), and the others similarly. First, x 2kj A () x 2 Rkj ðAÞ. Since Rkj ðAÞ is j-near rough set contained in A, then



No. of Pages 25

20


653

655

k k Rj ðAÞ \ A Rj ðAÞ Rkj ðAÞ ¼ Rkj ðAÞ ¼ 1: k

656

657 658 659

k

Thus 1 2 WAj ðxÞ and accordingly lAj ðxÞ ¼ 1. h

Remark 6.3. The j-near rough membership functions divide the universe U by using the j-near boundary, j-near positive and j-near negative regions of A # U, respectively into the following regions:

660

k

k

Bkj ðAÞ ¼ fx 2 Uj0 < lAj ðxÞ < 1g; POSkj ðAÞ ¼ fx 2 UjlAj ðxÞ ¼ 1g

662 663 664 665

666 667

k

and NEGkj ðAÞ ¼ fx 2 UjlAj ðxÞ ¼ 0g:

The following result is very interesting since it gives the relation between the rough j-membership functions and j-near rough membership functions. Moreover, it illustrates the importance of j-near rough membership functions. Lemma 6.3. Let hU; R; Cn i be Gn – CAS and A; B # U. Then, for each j 2 fr; l; i; ug and k 2 fp; s; cg, the following is true in general:

668

k

669 670

(i) lAj ðxÞ ¼ 1 ) lAj ðxÞ ¼ 1; k (ii) lAj ðxÞ ¼ 0 ) lAj ðxÞ ¼ 0;

8x 2 U . 8x 2 U .

671

672

673 674

Proof. (i) If lAj ðxÞ ¼ 1, then N j ðxÞ # A; 8x 2 U . Thus x 2 Rj ðAÞ and this implies x 2 Rkj ðAÞ which is a j-near rough set contained in A. Accordingly, k

675 676 677

lAj ðxÞ ¼ 1; 8x 2 U . (ii) If lAj ðxÞ ¼ 0, then N j ðxÞ \ A ¼ £; that contains x.

8x 2 U . But N j ðxÞ is a j-near rough set

678

k

k

Accordingly 0 2 KAj ðxÞ and this means that minðWAj ðxÞÞ ¼ 0. Hence

679

k

680

681 682 683

lAj ðxÞ ¼ 0;

8x 2 U. h

Remark 6.4. (i) According to the above result and by using Proposition 6.1, we k can prove that lAj is more accurate than lAj , this means that: k

684 685

(1) If x 2 A ) lAj ðxÞ 6 lAj ðxÞ. k (2) If x R A ) lAj ðxÞ 6 lAj ðxÞ.



No. of Pages 25

Generalized covering approximation space and near concepts

21

686 687

(ii) The converse of Lemma 6.3 is not true in general.

688

The following example illustrates Remark 6.5.

689 690 691 693 694 695 696 697 698 699 700 701 702

Example 6.3. Suppose the triple hU; R; Cn i be Gn – CAS, where U ¼ fa; b; c; dg and R ¼ fða; 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 of j = r and k = s the other cases similarly as follows: First, we have Nr ðaÞ ¼ fa; bg ¼ Nr ðbÞ; Nr ðcÞ ¼ U; Nr ðdÞ ¼ fdg. Thus, we can get The family of all r-semi rough sets is: Sr ðUÞ ¼ fU; £; fag; fdg; fa; bg; fa; dg; fa; cg; fc; dg; fa; b; cg; fa; b; dg; fa; c; dgg. Now consider the subset A ¼ fa; cg, then the r-rough membership functions of A; x 2 U are jfag \ Aj jfa; bg \ Aj 1 ¼ 1; lrA ðbÞ ¼ ¼ ; jfagj jfa; bgj 2 jfa; c; dg \ Aj 2 jfdg \ Aj lrA ðcÞ ¼ ¼ and lrA ðdÞ ¼ ¼ 0: jfa; c; dj 3 jfdgj lrA ðaÞ ¼

704 705 706

708 709

711

But the r-semi rough membership functions of A; x 2 U are jfag \ Aj jfa; bg \ Aj 1 sr WA ðaÞ ¼ ¼ 1; ¼ ; . . . ) lsAr ðaÞ ¼ 1: jfagj jfa; bgj 2 jfa; bg \ Aj 1 jfa; b; cg \ Aj 2 jfa; b; dg \ Aj 1 1 WsAr ðbÞ ¼ ) lsAr ðbÞ ¼ : ¼ ; ¼ ; ¼ jfa; bgj 2 jfa; b; cgj 3 jfa; b; dgj 3 3

712

714 715

717

jfa; cg \ Aj jfc; dg \ Aj 1 ¼ ¼ 1; ¼ ; . . . ) lsAr ðcÞ ¼ 1: jfa; cgj jfc; dgj 2 jfdg \ Aj jfa; dg \ Aj 1 sr WA ðdÞ ¼ ¼ 0; ¼ ; . . . ) lsAr ðaÞ ¼ 0: jfdgj jfa; dgj 2

WsAr ðcÞ

k

718 719

720 721 722

The j-near rough membership functions lAj allow us to define twelve different types of fuzzy sets in Gn – CAS as the following definition illustrates. Definition 6.3. Let hU; R; Cj i be Gn – CAS and A # U. Then for each j 2 fr; l; i; ug and k 2 fp; s; cg, the j-near fuzzy set in U is a set of ordered pairs: k A~kj ¼ fðx; lAj ðxÞÞjx 2 Ug.



No. of Pages 25

22 723 724 725

727 728

729

730 731 732 733 734 735 736 737 738 739 740 741

742 743 744 745 746 747 748 749 750


Example 6.4. According to Example 6.3, the r-semi fuzzy set of a subset A ¼ fa; cg is 1 s ~ Ar ¼ ða; 1Þ; b; ; ðc; 1Þ; ðd; 0Þ : 3 But the r-fuzzy set of a subset A ¼ fa; cg is A~r ¼ ða; 1Þ; b; 12 ; c; 23 ; ðd; 0Þ . 7. 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 second example, we apply our approaches in a multi-valued information system (MVIS) [14]. This type represents a generalization to information system because it 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 our jrough membership function in the case of j = r only. However, the other types of our j-rough membership functions are more accurate than j = r, so we can see that our approaches are the appropriate tools for these types. Example 7.1. Consider the following information system as in Table 7.1. Suppose we are given data about 6 students, as shown below: From Table 7.1, we have The set of universe: U ¼ f1; 2; 3; 4; 5; 6g. The set of attributes: AT ¼ fAnalysis; Algebra; Statisticsg ¼ C[ fDecisiong ¼ D. The sets of values: VAnalysis ¼ fBad; Goodg, VAlgebra ¼ fBad; Goodg, VStatistics ¼ fBad; Medium; Goodg and VDecision ¼ fAccept; Rejectg. But we take the set of condition attributes, C ¼ fAnalysis; Algebra; Statisticsg.

Table 7.1

Decision system.

Student

Analysis

Algebra

Statistics

Decision

1 2 3 4 5 6

Bad Good Good Bad Good Bad

Good Bad Good Good Bad Good

Medium Medium Good Bad Medium Good

Accept Accept Accept Reject Reject Accept



No. of Pages 25


23

Thus we have: U=C ¼ ff1g; f2; 5g; f3g; f4g; f6gg and the set of r-pre rough set is Pr ðUÞ ¼ }ðUÞ. 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): jf1g \ Xj jf2; 5g \ Xj 1 ¼ 1; lCX ð2Þ ¼ ¼ ; jf1gj jf2; 5gj 2 jf3g \ Xj jf6g \ Xj ¼ 1 and lCX ð6Þ ¼ ¼ 1: lCX ð3Þ ¼ jf3gj jf6gj

lCX ð1Þ ¼

759 760 761

763 764

766 767

769 770

772 773 774 775

Our Definition (r-pre rough membership function): jf1g \ Aj jf1; 2g \ Aj p p ¼ 1; ¼ 1; . . . ) lXr ð1Þ ¼ 1; WXr ð1Þ ¼ jf1gj jf1; 2gj jf2g \ Aj jf2; 6g \ Aj pr p ¼ 1; ¼ 1; . . . ) lXr ð2Þ ¼ 1; WX ð2Þ ¼ jf2gj jf2; 6gj jf3g \ Aj jf3; 5g \ Aj 1 p p WXr ð3Þ ¼ ¼ 1; ¼ ; . . . ) lXr ð3Þ ¼ 1 and jf3gj jf3; 5gj 2 jf6g \ Aj jf6; 5g \ Aj 1 pr p WX ð6Þ ¼ ¼ 1; ¼ ; . . . ) lXr ð6Þ ¼ 1: jf6gj jf6; 5gj 2 Moreover, for some elements that has decision (Reject) such that 5 we get: ¼ 12, that is 5 may be belongs to the set XðDecision : In Pawlak: lCX ð5Þ ¼ jf2;5g\Xj jf2;5gj AcceptÞ, X ¼ f1; 2; 3; 6g and this contradicts to Table n 7.2. o p

776 777 778 779 780

But in our definition: we have WXr ð5Þ ¼

jf5g\Aj jf5gj

¼ 0; jf5;6g\Aj ¼ 12 ; . . . ) jf5;6gj

p

lXr ð5Þ ¼ 0. This means that 5 does not belongs to the set XðDecision : AcceptÞ ¼ f1; 2; 3; 6g which is coincide with Table 7.1. Hence, our approaches are more accurate than Pawlak definition.

Table 7.2 Person

Languages

Sports

Skills

Decision

A B C D E

{E} {E, G} {G} {G, A} {A}

{H} {H} {H, B} {B} {T}

{S} {S} {R} {R, F} {F}

Accept Reject Accept Accept Reject



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24 781 782 783 784 785 786 787 789


Example 7.2. Consider the following multi-valued information system (MVIS) as in Table 7.2. Suppose we are given data about 5 persons, as shown below: where R1 ¼ Languages ¼ fEnglish; German; Arabicg, R2 ¼ Sports ¼ fHandball; Basketball;Tennisg and R3 ¼ Skills ¼ fSwimming;Running; Fishingg. Such that xRn y; 8n ¼ 1; 2; 3: We shall use the case of j = r and k = c as follows: aR1 ¼ fa; bg;

bR1 ¼ fbg;

cR1 ¼ fb; c; dg;

dR1 ¼ fdg;

eR1 ¼ fd; eg;

790 792

aR2 ¼ fa;b;cg; bR2 ¼ fa; b; cg; cR2 ¼ fcg; dR2 ¼ fc; dg; eR2 ¼ feg and

793 795 796 797 798 799 800 801 802 803 804

806 807 808 809 810 811 812 813 814 815 817 816 818 819

821 822 825 823 824 827 828 826

aR3 ¼ fa; bg;

cR3 ¼ fc; dg;

dR3 ¼ fdg;

eR3 ¼ fd; eg:

In order to represent the set of all condition attributes, we generate the followT ing relation from all above relations as follows: xR ¼ 3n¼1 xRn . Thus we get aR ¼ fa; bg; bR ¼ fbg, cR ¼ fc; dg; dR ¼ fdg; eR ¼ feg. 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: Suppose that XðDecision : AcceptÞ ¼ fa; c; dg. Lin Definition [10] (rough membership function): lR X ðbÞ ¼

jfa;bg \ Xj jfc; dg \ Xj 1 jfdg \ Xj ¼ 1; lR ¼ and lR ¼ 1: X ðcÞ ¼ X ðdÞ ¼ jfa; bgj jfc; dgj 2 jfdgj

Our approaches ðGn – CASÞ. 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: Nr ðaÞ ¼ fa; bg, Nr ðbÞ ¼ fbg, Nr ðcÞ ¼ fc; dg, Nr ðdÞ ¼ fdg, Nr ðeÞ ¼ feg. The l-neighborhoods of all elements are: Nl ðaÞ ¼ fag, Nl ðbÞ ¼ fa; bg, Nl ðcÞ ¼ fcg, Nl ðdÞ ¼ fc; dg, Nl ðeÞ ¼ feg. The i-neighborhoods of all elements are: Ni ðaÞ ¼ fag, Ni ðbÞ ¼ fbg, Ni ðcÞ ¼ fcg, Ni ðdÞ ¼ fdg, Ni ðeÞ ¼ feg. 1. r-rough membership function: lrX ðaÞ ¼

jfa; bg \ Xj jfc;dg \ Xj 1 jfdg \ Xj ¼ 1; lrX ðcÞ ¼ ¼ and lrX ðdÞ ¼ ¼ 1: jfa; bgj jfc; dgj 2 jfdgj

2. l-rough membership function: llX ðaÞ ¼

830

bR3 ¼ fa; bg;

jfag \ Xj ¼ 1; jfagj

llX ðcÞ ¼

jfcg \ Xj ¼1 jfcgj

and llX ðdÞ ¼

jfc; dg \ Xj ¼ 1: jfc; dgj



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25

3. i-rough membership function: liX ðaÞ ¼

jfag \ Xj ¼ 1: jfagj

liX ðcÞ ¼

jfcg \ Xj ¼ 1: jfcgj

liX ðdÞ ¼

jfdg \ Xj ¼ 1: jfdgj

842

It is clear that Lin rough membership function is the same as r-rough membership function. Moreover, our approaches l-rough (resp. i-rough) membership function is more accurate than r-rough membership function and Lin rough membership function.

843

8. Conclusions and future works

839 840 841

853

In this work, we introduced one of an important topological application that named ‘‘near concepts’’ in rough context. Accordingly, different types of approximations (resp. rough membership relations and functions) 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 applications and problems.

854

References

844 845 846 847 848 849 850 851 852

855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878

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