Applications of Some Topological Near Open Sets to Knowledge Discovery

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1 IJACS International Journal of Advanced Computer Science Applications Vol 7 No Applications of Some Topological Near Open Sets to Knowledge Discovery A S Salama Tanta University; Shaqra University Tanta Egypt; Sajir KSA O G El-Barbary Shaqra University Faculty of Science Tanta University Sajir KSA Abstract In this paper we use some topological near open sets to introduce the rough set concepts such as near open lower near open upper approximations Also we study the concept of near open rough set some of their basic properties We will study the comparison among near open concepts rough set concepts We also studied the effect of this concept to motivate the knowledge discovery processing Keywords Topological spaces; ough sets; Knowledge discovery; open sets; Accuracy measure I INTODUCTION Information technology is huge need more accurate measures to discover their valuable knwoledge An important field of this technology is the information discovery via available information ough set theory their topological generalizations using some topological near open sets [ ] are a recent an accurate applicable approach for reasoning about data Many researchers generalized rough sets but the concept of a topological rough set given by Wiweger [16] in 1989 is the basic starting point This generalization is basically starting with a topological space defined the approximation via the interior the closure operators on topological spaces The concept of -open set was introduced in 1983 by M E Abd El-Monsef others [3] In this paper we introduce investigate the concept of near open approximation space These spaces help to get a new classification for the universe Also we investigate the concept of near open lower near open upper approximations We study near open rough sets the comparison between this concept rough sets is also studied Also we give some counter examples II OUGH SET BASIC CONCEPTS Classical rough set theory has come from the need to represent subsets of a universe in terms of equivalence classes of a partition of that universe The partition characterizes a topological space called approximation space K = X ) where X is a set called the universe is an equivalence relation [8 14] The equivalence classes of are also known as the granules elementary sets or blocks; we will use x X to denote the equivalence class containing x X In the approximation space we consider two sets namely = { x X : x A = / φ} = { x X : x A} that is called the lower the upper approximation of A X respectively Also let POS = denote the positive region of A NEG = X denotes the negative region of A BN = denote the boundary region of X The degree of completeness can also be characterized by the accuracy measure in which represents the cardinality of set as follows: α = where A = / φ Accuracy measures try to express the degree of completeness of knowledge α is able to imprisonment how large the boundary region of the data sets is; however we cannot easily capture the structure of the knowledge A fundamental advantage of rough set theory is the ability to hle a category that cannot be sharply defined given a knowledge base Characteristics of the potential data sets can be measured through the rough sets framework We can measure inexactness express topological characterization of imprecision with: 1) If = φ = / X then A is roughly - definable 2) If = φ = / X then A is internally - undefinable 3) If = / φ = X then A is externally - undefinable 4) If = φ = X then A is totally - undefinable We denote the set of all roughly -definable resp Internally -undefinable externally -undefinable totally -undefinable) sets by D resp IUD / EUD TUD ) wwwijacsathesaiorg 367 P age

2 IJACS International Journal of Advanced Computer Science Applications Vol 7 No With α A ) classifications above we can characterize rough sets by their size of the boundary region A topological space [6] is a pair X τ ) consisting of a set X family τ of subsets of X satisfying the following conditions: 1) φ X τ 2) τ is closed under arbitrary union 3) τ is closed under finite intersection τ The pair is called a topological space the elements of X are called points of the space the subsets of X belonging to are called open set in the space the complement of the subsets of X belonging to τ are called closed set in the space; the family τ of open subsets of X is also called a topology for X A = { F X : A F F is close is A X called τ -closure of a subset Evidently A is the smallest closed subset of X which contains A Note that A is closed iff A = A A = { G X : G A G called the τ -interior of a subset A X Evidently is open} A is the union of all open subsets of X A = A which containing in A Note that A is open iff And b = A A is called the τ -boundary of a subset A X Let A be a subset of a topological space X τ ) Let A A b be closure interior boundary of A respectively A is exact if b = φ otherwise A is rough A = A A X has two possibilities rough or exact It is clear A is exact iff is In Pawlak space a subset Definition 21 [3] A subset A of a topological space open X τ ) is called if A cl int cl )) The complement of -open set is -closed set We denote that the set of all open -closed sets by O C respectively emark 21 For any topological space X τ ) We have τ O III TOPOLOGICAL NEA OPEN OUGH CLASSIFICATION In this section we introduce investigate the concept of near open approximation space Also we introduce the concepts of near open lower approximation near open upper approximation study their properties Definition 31 Let X be a finite non-empty universe The pair X ) is called a near open approximation space where is a general relation used to get a subbase for a topology τ on X a class of -open sets O emark 31 In Definition 31 we use the symbol to avoid confusion with which refers to an equivalence relation Example 31 Let U = { a d e f g} be a universe of multi valued information system of seven patients A B C are conditional attributes D is the decision attribute) as in Table 1 the relation defined on patients by = { a a) a c) a d) b b) d) c a) b) c d) d a)} thus a = { a b = { c = { a d = { a} Then the topology associated with this relation is τ = { U φ { a} { d }{ a { b { a { a } O = { X φ { a} { d }{ a { a { b { { a { a { } So U ) is a near open approximation space U a b c d e f g TABLE I A {A2} {A1A2} {A2} {A1} {A1} {A1} {A1} MULTIVALUED INFOMATION SYSTEM B {B1B2B4} {B1B2} {B1B2} {B1B2B4} {B5} {B1B2} {B1B3B4} C {c1} {c1c3} {c1} {c4} {c1c2} {c1} {c1c3} D {d1} {d2} Example 32 Let X = { a be a subset of the universe U a relation defined on X by a = { a b = { c = { a Then the subtopology associated with this relation is τ = { X φ { b }{ a } O = { X φ { { a { } So X ) is a subnear open approximation space Definition 32 Let X approximation space near open lower resp near open upper) approximation of any non-empty subset A of X is defined as: = { G O : G A} = { F C : F A} Definition 33 Let X approximation space From the relation wwwijacsathesaiorg 368 P age

3 IJACS International Journal of Advanced Computer Science Applications Vol 7 No int int A cl cl A X with respect to any A X as shown in Figure 1 for any The Universe X can be divided into 12 regions Fig 1 egions of the universe emark 32 The study of near open approximation spaces is a generalization for the study of approximation spaces Because of the elements of the regions [ ] will be defined well in A while this point was undefinable in Pawlak s approximation spaces Also the elements of the region [ ] do not belong to A while these elements was not well defined in Pawlak s approximation spaces In our study reduce the boundary of A in Pawlak s approximation space by near open boundary of A Also we extend exterior of A which contains the elements did not belong to A by near open exterior of A Proposition 31 For any near open approximation space X ) the following are hold of any A X : 1) = Edg Edg b Edg A Edg = A 2) b = Edg Edg = Edg = A Edg = A Proof 2) It follows from b = = A ) = Edg Edg Proposition 32 For any near open approximation space X ) the following are hold of any A X : = Edg A Edg A 1) ) ) 2) = Edg Edg Proof Obvious Proposition 33 For any near open approximation space X ) The following are hold to any A X : 1) Edg = Edg ) 2) Edg = Edg ) Proof Obvious Definition 34 Let X approximation space A X Then there are memberships say near open strong near open weak memberships respectively which defined by: 1) x A iff x 2) x A iff x 3) x A iff x 4) x A iff x emark 33 According to Definition 34 near open lower near open upper approximations of a set A X can be written as 1) = { x A: x A} 2) = { x A: x A} emark 34 Let X approximation space A X 1) x A x A Then 2) x A x A The converse of emark 34 may not be true in general as seen in the following example X = { a Example 33 Let be a universe a relation = { a a) d c) d d) defined by a) d) c)} a = { a} b = φ thus c = { a d = { Then the topology τ = { X φ { a} associated with this relation is { { a } So X ) is a -approximation A = { space Let we have b A but b / A Also B = { let We have d B but / B d We can characterize the degree of completeness by a new tool named -accuracy measure defined as follows: wwwijacsathesaiorg 369 P age

4 IJACS International Journal of Advanced Computer Science Applications Vol 7 No α = where A = / φ Example 34 According to Example 31 we can deduce Table 2 below that show the degree of accuracy measure α -accuracy measure α for some proper subsets of the universe of discourse TABLE II ACCUACY MEASUES A X α α {a} 1/2 1 {a 1/2 1 { 1/3 1 { 2/3 1 We see that the degree of exactness of the set A = { a} by using accuracy measure equal to 5% by using near open accuracy measure equal to 1% Consequently near open accuracy measure is the accurate measure in this case We investigate near open rough equality near open rough inclusion based on rough equality inclusion which introduced by Pawlak Novotny in [1 11] Definition 35 Let X approximation space B are: A B X Then we say that A i) Near open roughly bottom equal if = ii) Near open roughly top equal if = iii) Near open roughly equal A if = = Example 35 According to Example 31 we have the sets { a { a are near open roughly bottom equal { c { b are near open roughly top equal One can easily show that is an equivalence relation on P hence the pair P ) is an approximation space Also this relation is called near open rough equality of the near open approximation space X ) Definition 36 Let X approximation space We define the equivalence relation on the set P given by the condition: E A E if - int = - int - cl = - cl The equivalence relation E is precisely the same of since = int = cl emark 35 For any subset A of X the equivalence class of the relation or E ) containing A is denoted by [A] or [ A] EB Thus [ A] = { D X : D) = D) = } We denote by of the family of near open rough classes of a near open approximation space X ) Definition 37 Let X approximation space A B X Then we say that i) A is near open roughly bottom included in B if ii) A is near open roughly top included in B if iii) A is near open roughly included in B if Example 36 According to Example 31 we have { a is near open roughly bottom included in { a Also { c is near open roughly top included in { b IV POPETIES OF NEA OPEN OUGH SETS In this section we introduced a new concept of near open rough set Definition 41 For any near open approximation space X ) a subset A of X is called: 1) -definable -exact)if = b = φ 2) near open rough if = / or b = / φ Example 41 Let ) space as in Example 33 We have the set { or X be a near open approximation a is near open exact while { is near open rough set Proposition 41 Let X approximation space Then: 1) Every exact set in X is near open exact 2) Every near open rough set in X is rough Proof Obvious The converse of all parts of Proposition 42 may not be true in general as seen in the following example wwwijacsathesaiorg 37 P age

5 IJACS International Journal of Advanced Computer Science Applications Vol 7 No Example 42 Let X approximation space as in Example 33 Then the set { a is near open exact but not exact the set { a is rough but not near open rough emark 41 The intersection of two near open exact sets need not be near open exact set Example 43 Let X approximation space as in Example 33 We have { b { a are two near open exact sets but { { a = { does not near open exact Definition 42 Let X approximation space the set 1) oughly -definable if A X is called: / φ = = / X 2) Internally -undefinable if = φ = / X 3) Externally -undefinable if = / φ = X 4) Totally -undefinable if = φ = X We denote the set of all roughly -definable resp Internally -undefinable externally -undefinable totally -undefinable) sets by D resp IUD EUD TUD ) emark 42 For any near open approximation space X ) The following are held: 1) D D 2) IUD IUD 3) EUD EUD 4) TUD TUD Example 44 According to Example 33 we have the set { a D { IUD { a EUD but but but { a / D The set { IUD / Also the set { a EUD Lemma 41 For any near open approximation space X ) for all x y X the condition x {y}) y {x}) implies { x}) = { y}) Proof By definition of near open upper approximation of a set is a -closure of this set since cl{y}) is a - closed set containing x by the condition) while cl{x}) is the smallest -closed set containing x thus cl{ x}) cl{ y}) Hence { x}) { y}) The opposite inclusion follows by symmetry cl{ y}) cl{ x}) Hence { y}) { x}) which complete the proof Lemma 42 Let X approximation space which every -open subset A of X is -closed Then y {x}) implies x {y}) for all x y X Proof If x / {y}) then there exists a -open set G containing x G {y} = φ such that which implies that { y} X \ G) X \ G) but is a -closed set also is a -open set does not contain x thus X \ G) { x} = φ Hence y / {x}) Proposition 42 Let X approximation space every open subset A of X is -closed Then the family of sets { { x}) : x A } is a partition of the set X Proof If x y z A z { x}) { y}) then z {x}) z {y}) Thus by Lemma 42 x {z}) y {z}) by Lemma 41 we have { x}) = { z}) { y}) = { z}) Therefore { x}) = { y}) = { z}) Hence either { x}) = { y}) or { x}) { y}) = φ V POPETIES OF NEA OPEN APPOXIMATION SPACES The purpose of this section is to investigate some properties of near open approximation spaces Proposition 51 Let X approximation space A B X Then A A i) ) ii) iii) If φ) = φ) φ = = X = A B then Proof i) Let x which mean that wwwijacsathesaiorg 371 P age

6 IJACS International Journal of Advanced Computer Science Applications Vol 7 No x { G O G A} Then there exists G O such that x G A Thus x A Hence A Also let x X by definition of = { F C A F} then x F for all F C Hence A ii) Follows directly iii) Let x by definition of near open lower approximation of A we have x { G O G A} A thus G B but B x G then x Also let x / this means that x / { F C B F} then there exists F C B F x / F which means that there F CX A B F x / F which x / { F C A F thus x / exists ) implies } Therefore Proposition 52 Let X approximation space A B X Then i) X \ = X \ ii) X \ = X \ iii) ) = iv) ) = v) ) ) vi) ) ) Proof i) Let x X \ which is equivalent to x { G O G X \ A} So there exists G O ) such that x G X \ A X exists c G such that c A G Then there c c x / G G C ) X Thus x / So x X \ Therefore X \ = X \ ii) Similar to i) iii) Since = { G O G A} This implies that ) = { { G O G A}} = { G O G A} = iv) X \ iii) we get ) = X \ X \ X \ ) X \ ) = From i) ii) ) = X \ X \ = X \ X \ ) = A v) Since )) by iii) we have ) = then ) ) A A vi) Since )) ) by iv) we have ) = then ) ) Proposition 53 Let X approximation space A B X Then i) A ii) A iii) A iv) A Proof i) Since we have A A B B A B Then A A by iii) in Proposition 51 then A ii) iii) iv) Similar to i) The equality of all parts in Proposition 53 are not held as shown in the following example Example 51 According to Example 31: i) If A = { B = { a then we have A = { a = { = { a} Therefore A = / ii) If = { A = { a B then we have A = X = { = { a Therefore = / A iii) If A = { a B = { then we have A = φ = { a = { Therefore A = / iv) If = { A B = { then we have wwwijacsathesaiorg 372 P age

7 IJACS International Journal of Advanced Computer Science Applications Vol 7 No A = φ = { = { Therefore = / A The following theorems are a generalization of Proposition 53 Theorem 51 Let X approximation space A B X If A is -definable Then the following are held i) A = ii) A = Proof i) It is clear that A For the converse inclusion let x A that means x { G O G A B} Then there exists G O such that x G A B We distinguish three cases: Case 1) If G A x G G is a -open set then x G A = φ then G B x G Case 2) If thus x Case 3) If G A = / φ Since x G G is a - open set then x cl for every G which has the above condition thus thus x then x because A is - definable Hence in three cases x ii) It is clear that A We prove the converse inclusion Let x then x implies x x G X where G is a -open set x implies for all G O G B =/ φ Therefore G A = G B = G Y = / φ Hence x A Theorem 52 Let X approximation space A B X Then the following are held i) cl = cl ii) int = int Proof i) By Proposition 51 i) Proposition 53 ii) we have cl cl ) Then cl cl ) On the other cl h since cl B cl the union of a -open set a closed set is -closed then cl cl ) = cl Therefore cl = cl ii) Since the intersection of an open set int a - open set is -open int = int ) int On the other h by using Proposition 53 iii) int int ) int Therefore int = int VI CONCLUSIONS In this paper we used general binary relations to generate rough approximations Properties of rough belonging rough equality rough power set are now clear with respect to any relation type raised from any field of applications We have proved that there exist similar properties there exist different properties among the approximations raised from this paper that studied from many authors [ ] We will investigate new applications of these tools using topological generalizations in a future paper EFEENCES [1] H M Abu-Donia "Multi knowledge based rough approximations applications" Knowledge-Based Systems ) 2-29 [2] H M Abu-Donia A S Salama "Generalization of Pawlak's rough approximation spaces by using δ-open sets" International Journal of Approximate easoning ) [3] D G Chen W X Zhang "ough sets topological spaces" Journal of Xian Jiaotong University 35 21) [4] A Galton "A generalized topological view of motion in discrete space" Theoretical Computer Science ) 23) [5] J L Kelley "General Topology" Van Nostr Company 1995 [6] M Kondo "On the structure of generalized rough sets" Information Sciences ) [7] J Kortelainen "On the relationship between modified sets topological spaces rough sets" Fuzzy Sets Systems ) [8] M Kryszkiewicz "ough set approach to incomplete information systems" Information Sciences ) [9] M Kryszkiewicz "ules in incomplete information systems" Information Sciences ) [1] E F Lashin A M Kozae A A Abo Khadra T Medhat "ough set theory for topological spaces" International Journal of Approximate easoning 4 25) [11] C Largeron S Bonnevay "A pre topological approach for structure analysis" Information Sciences ) [12] Zhaowen Li Tusheng Xie Qingguo Li "Topological a structure of generalized rough sets" Computers Mathematics with Applications 63 6) 212) [13] T Y Lin "Granular fuzzy sets: A view from rough set probability theories" International Journal of Fuzzy Systems 3 2) 21) wwwijacsathesaiorg 373 P age

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