Minimal structure approximation space and some of its application

1 Introduction

Popa and Noiri [18] introduced the notion of minimal structure. Also, the definitions and characterizations of separation axioms by using the concepts of a minimal structure are introduced in [14, 19]. Moreover, some new types of sets on the minimal structure and separation axioms for generalized topology and study some of its properties and applications are investigated in [1, 15]. In 1982 rough set theory has been introduced by Pawlak [12] as an extension of set theory and defined the approximation space (U . R), where U is a universal set and R is an equivalence relation. Also, he introduced the notions of the upper approximation U (A), lower approximation L (A), and the boundary approximation b (A) of any subset A of U in [16, 17]. One of the most important results in rough sets is L (A) ⊆ A ⊆ U (A) and b (A) = U (A) - L (A). The authors in [7, 9] discussed the relation between the rough sets, fuzzy sets, information systems, and some of its applications. Also, in [2, 20] the authors tried to generalize these concepts on the general relations in several different ways, for example, of these methods is to use the concept of the neighbourhood of the element. But, some of the contradictions have emerged with the results of Pawlak the most prominent of these contradictions is L (A) ⊈ A ⊈ U (A). To overcome this contradiction, we will use the concept of minimal structure. In this paper, we introduce a new space called a minimal structure approximation space, which is based on a binary relation R on the generalized neighborhood system on a nonempty set by using a minimal structure and study some of its properties. Space helps to get a new classification for definability. This classification depends on the topological concepts in minimal topological space. Our classification is more related to real-life problems. Some properties of the approximation in MSAS are investigated. Also, we define the lower and upper approximations of the space. Finally, we use the separation properties in the process of approximations and definability. This article opens the way for future use of separation axioms in the application.

Definition 1.1. [16] Let U be a nonempty finite set and R be an arbitrary binary relation on U. Then (U, R) is called the generalized approximation space.

Definition 1.2. [20] For an approximation space (U, R) The right neighborhood of an element x ∈ U is N (x) = {y ∈ U : xRy} And the collection of all a right neighborhoods in (U, R) is N (U) = {N (x) : x ∈ U}.

Definition 1.3. [18] A subfamily A of P ( U ) where P ( U ) the family of all subsets of U is called a minimal structure on U(briefly MS(U)), if φ and U belong to A .

2 Minimal structure approximation space

In this section, we use a minimal neighborhood to define the minimal structure approximation space (MSAS). Also, we investigate a minimal lower, a minimal upper approximation, MS - undefinable (rough) set and MS - internally, MS - externally and MS - totally.

Definition 2.1. Let (U, R) be an approximation space. We define a minimal structure (U, R) as MS (U) = {φ, U} ⋃ N (U) where, N (U) = {N (x) : x ∈ U}.

The members of the minimal structure MS (U) are called MS-open sets and the triple (U, R, MS) is a minimal structure approximation space in short MSAS. The complement of an MS-open set is called MS-closed set, and the class of all MS-closed sets denoted by (MS (U))  ^c and (MS (U))  ^c  = {A ^c  : A ^c  ∈ MS (U)} where A ^c is the complement of A with U.

Example 2.2. Let U = {a, b, c, d} and R = {(a, c) , (a, d) , (c, c) , (b, d) , (c, b) , (d, a)}. Then the neighborhoods of elements x ∈ U are N (a) = {c, d} , N (b) = {d} , N (c) = {b, c}, N (d) = {a} and the minimal structure set MS (U) = {∅ , {a} , {d} , {b, c} , {c, d} , U} and (MS (U)  ^c  = {∅ , {a, b} , {a, d} , {a, b, c} , {b, c, d} , U}.

Definition 2.3. Let (U, R, MS) be MSAS and A ⊆ U. Then A is called MS- definable (exact) if and only if MS (A) = (MS (A))  ^c .

Definition 2.4. [18] A minimal structure MS (U) called a generalized topology on U if it is closed under unions in the sense that B ⊆ MS ( U ) implies ⋃ B ∈ MS ( U ) .

Definition 2.5. For a MSAS (U, R, MS). We define

A minimal lower approximation of A ⊆ U, for short LMS (A) is LMS (A) = ⋃ {V :   V ∈ MS (U) ,   V ⊆ A}

Theorem 2.6. For a MSAS (U, R, MS) and A ⊆ U. We have x ∈ LMS (A) if and only if for every MS - openG and x ∈ G, G ⋂ A ≠ φ.

Proof. Follows from Definition 2.5.□

Definition 2.7. [6] For a MSAS (U, R, MS) if the union of any family of elements of MS (U) is in MS (U). Then MS (U) has property of Maki.

Theorem 2.8. For a MSAS (U, R, MS), A ⊆ U and MS (U) has a property of maki. Then the following properties holds:

LMS (A) = A if and only if A ∈ MS (U);

Proof. Follows from the definitions of MS-closed set, LMS (A) and UMS (A).□

Definition 2.9. For a MSAS (U, R, MS) and A ⊆ U. Then

A minimal boundary of A (MS - b (A)) is MS - b (A) = UMS (A) - LMS (A);

Proposition 2.10. For a MSAS (U, R, MS) and A ⊆ U. Then A is called MS-definable (exact) iff UMS (A) = LMS (A) .

Proof. Follows from the Definition 2.3 and Definition 2.5. □

Example 2.11. Let U ={ a, b, c, d } and R be a binary relation on U, and N (a) ={ a, b }, N (b) ={ b, c }, N (c) ={ d }, N (d) ={ c }. Then MS (U) = {∅ , {c} , {d} , {a, b} , {b, c} , U} and MS (U)  ^c  ={ ∅ , { c, d } , { a, d } , { a, b, d } , { a, b, c } , U }. Let A ={ c, d } , B = { b, c } , C = { b, c, d }. Then LMS (A) = {c, d} = A = UMS (A), hence MS - b (A) =∅ and A is MS - definable set.

For the set B, we have LMS (B) = B, UMS (B) ={ a, b, c }. Then MS - b (B) = {a}, LMS - edg (B) =∅ and UMS - edg (B) = { a } .

Finally, for the set C, we have LMS (C) ={ c, d } and UMS (C) = U.Then MS - b (C) ={ a, b }, LMS - edg (Z) ={ a } and UMS - edg (C) = { b } .

The relationship between the minimal external edge, minimal internal edge and minimal boundary edge are given by proposition 2.12.

Proposition 2.12. For a MSAS (U, R, MS) and A ⊆ U. Then MS - b (A) = LMS - edg (A) ∪ UMS - edg (A)

Proof. Obvious. □

Proposition 2.13. For a MSAS (U, R, MS) and A ⊆ U. Then A is called MS - definable iff MS - b (A) =∅

Proof.Follows directly from Definition 2.9 and Proposition 2.10. □

Definition 2.14. For a MSAS (U, R, MS) and A ⊆ U. Then A is called

MS-internally definable, iff LMS (A) = A,

From the topological interpretation of LMS (A) and UMS (A), we reformulate the preceding definition by using the topological concept.

Definition 2.15. Let (U, R, MS) be MSAS and be a minimal topological space. Then for every A ⊆ U, we have:

A is called MS-internally (MS-externally, MS-totally) definable iff A is minimal open (minimal closed, minimal clopen) set in a minimal topological space,

Proposition 2.16. For a MSAS For a MSAS (U, R, MS) and A ⊆ U. Then A is MS- definable if it is MS- internally definable and MS-externally definable.

Proof. Directly from Proposition 2.10 and Definition 2.14. □

3 Some properties and applications of Minimal structure approximation space

The aim is to investigate some properties of approximation in MSAS. Also, we focused on the deviations between (MSAS), neighborhood approximation space and Pawlak approximation.

Proposition 3.1. For a MSAS (U, R, MS) and A ⊆ U. Then

LMS (A) ⊆ A,

Proof. 1- Let a ∈ LMS (A) hence there exists G ∈ MS (U) such that a ∈ G ⊆ A. Therefor a ∈ A 2- Let a ∈ A and F ∈ (MS (A))  ^c with A ⊆ F. Then a ∈ F and a ∈ UMS (A).

3 and 4 are follow directly.

5- Let a ∈ LMS (A). Therefor G ∈ MS (A) such that a ∈ G ⊆ A. Also, if A ⊆ B, then a ∈ G ⊆ B and G ∈ LMS (B). To prove UMS (A) ⊆ UMS (B) let a ∉ UMS (B). Therefor F ∈ (MS (A))  ^c  : B ⊆ F and a ∉ F. If A ⊆ B, then a ∉ UMS (A). □ In proposition 3.2, we introduce the property which is essential in the study of some concepts in MSAS as well as Rough set concepts.

Proposition 3.2. For a MSAS (U, R, MS) and A ⊆ U. Then

LMS (A ^c ) = (UMS (A))  ^c ;

Proof.

Since UMS (A ^c ) = ⋂ {F : F ∈ (MS (A))  ^c , A ^c  ⊆ F} = ⋂ {U - G : G ∈ MS (A) , A ^c  ⊆ U - G} = U -∪ { G ∈ MS (A) : G ⊆ A }= U - LMS (A) = (LMS (A))  ^c .

Proposition 3.3. For a MSAS (U, R, MS) and A ⊆ U. Then

A is MS-internally definable if and only if A ^c is MS -externally definable;

Proof. Follows from Definition 2.3. □

Proposition 3.4. For a MSAS (U, R, MS) and A ⊆ U. Then

LMS (A ⋃ B) ⊇ LMS (A) ⋃ LMS (B);

Proof.

Since A ⊆ (A ⋃ B), B ⊆ (A ⋃ B), LMS (A) ⊆ LMS (A ⋃ B)   and LMS (B) ⊆ LMS (A ⋃ B). Then LMS (A) ⋃ LMS (B) ⊆ LMS (A ⋃ B).

The connection for minimal lower and minimal upper operations between a subset A and the complement is given in the following proposition.

Proposition 3.5. For a MSAS (U, R, MS) and A ⊆ U. Then

UMS (A) ⋃  UMS (A ^c ) = U;

Proof.

Since A ⊆ UMS (A), A ^c  ⊆ UMS (A ^c ). and U = A ∪ (A ^c ) ⊆ UMS (A) ∪  UMS (A ^c ). Then UMS (A) ⋃  UMS (A ^c ) = U

Proposition 3.6. Let U be a nonempty universe and R ⊆ U ×  U be preordering relation (reflexive and transitive) on U. Then MS (U) is a base for topology on U.

Proof. Let Ω ={ N (A) : A ∈ U } be the set of all right neighborhoods of all elements x ∈ U. If x ∈ N (x)   ∀ x ∈ U and R is reflexive, then U = ⋃ _x∈UN (x). Let N (x) , N (y) ∈ Ω, N (x)∩ N (y) ≠ ∅ for all x, y ∈ U. If R is transitive, then their exists z ∈ U such that z ∈ N (x) and z ∈ N (y), but z ∈ N (x) implies N (z) ⊆ N (x) and z ∈ N (y) implies N (z) ⊆ N (y). Then N (z) ⊆ N (x) ∩ N (y). Then Ω is a base for topology on U. So MS (U) is a base for topology on U. □

Example 3.7. Let U ={ a, b, c } and R = {(a, a) , (b, b) , (c, c) , (b, a) , (c, a) , (b, c)}. Then N (a) ={ a } , N (b) = U, N (c) = { a, c }, MS (U) ={ ∅ , { a } , { a, c } , U } is a base for a topology on U.

Proposition 3.8. For a MSAS (U, R, MS) and R is preordering relation on U. Then:

LMS (A ⋂ B) = LMS (A) ⋂ LMS (B) ,

Proof.

By Proposition 3.4, we have LMS (A ⋂ B) ⊆ LMS (A) ⋂ LMS (B). Since LMS (A) ⊆ A and LMS (B) ⊆ B. Then LMS (A) ⋂ LMS (B) ⊆ A ⋂ B. But LMS (A ⋂ B) is largest minimal structure contained in A ⋂ B. Then LMS (A) ⋂ LMS (B) ⊆ LMS (A ⋂ B). Hence LMS (A ⋂ B) = LMS (A) ⋂ LMS (B)  .

Proposition 3.9. For a MSAS (U, R, MS), A ⊆ U and R is an equivalence relation. Then

LMS (LMS (A)) = UMS (LMS (A)),

Proof. If R is an equivalence relation, then MS (U) is a quasi-discrete topology (every open set is closed set). These complete the proof. □

Example 3.10. Let U = {dye₁, dye₂, dye₃, . . . , dye₅₀} be the set of 50 Chemical dyes and A={HUMO, LUMO, E, Total Energy, Dipole Moment, Hardness (η), Softness (σ), Electronegativity (X), Mechanical potential(μ), Electrophilicty(W), IPCE} be the set of attribute as in Table 1 (see Figure 1). By using table 1, we generate Table 2 (see Figure 2) and the relation between dyes as (dye _i , dye _j ) ∈ R for i, j ∈ {1, 2, . . . , 50} if the value of dye _i equal value of dye _j for any attribute in A in table 2 to generate the MSAS for example for attribute HUMO as MSAS (HUNO) = {dye {1, 11, 19, 21, 22, 23, 49, 50} , dye {16, 17, 25, 36, 45, 47}, dye {13, 15, 43, 44} , dye {2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 26, 28, 31, 32, 33, 37, 38, 39, 40, 42} , dye {3, 18, 20, 24, 28, 29, 30, 35, 41, 46, 48}}. And by similar way we can find MSAS for al attribute. Also by using the same role of (Lashin et al., 2005) for reduct and core of knowledge and by generating the MSAS, we find the reduct of knowledge as in table 3. And the core is the set HUMO, Total Energy, Dipole Moment

OPEN IN VIEWER

Fig. 1

Table 1 (Table of Data).

OPEN IN VIEWER

Fig. 2

Table 2 (Table of Coding).

Separation axioms play an important role to separate points and sets in a topological space. The main purpose is to direct the attention to the importance of separation properties as methods for isolating individuals in a given collection data via their information.

Definition 4.1. A minimal structure approximation space (U, R, MS) is called:

T₀-space iff for every two distinct points x and y in U there exist MS-open set G such that x ∈ G and y ∉ G,

Clearly T₂ ⇒ T₁  ⇒ T₀

Proposition 4.2. A minimal structure approximation space (U, R, MS) is T₀ if and only if UMS ({ x }) ≠ UMS ({ y }) for all x ≠ y, x, y ∈ U .

Proof. Let x ≠ y. Then there exists G ∈ MS (U) such that x ∈ G and y ∉ G. First, there exists U - G ∈ (MS (U))  ^c ,   x ∉ U - G, y ∈ U - G if and only if A ⋂ {U - G : U - G ∈ (MS (U))  ^c , {y} ⊆ G} if and only if x ∉ UMS ({ y }) but x ∈ UMS ({ x }). Then UMS ({ x }) ≠ UMS ({ y }) . Second, let UMS ({ x }) ≠ UMS ({ y }) i.e, there exists z ∈ U such that z  ∈ UMS ({ x }) and z ∉ UMS ({ y }) . Suppose x ∈ UMS ({ y }) implies UMS ({ x }) ⊆ UMS ({ y }) implies z ∈ UMS ({ y }).These give contradiction. So x ∉ UMS ({ y }). Hence U - UMS ({ y }) is a minimal structure contains x nor y.Then MSASis T₀-space. □

Proposition 4.3. A minimal structure approximation space (U, R, MS) is T₁-space if and only if {x} is MS-externally definable, for all x ∈ U.

Proof. Let x ∈ U and (U, R, MS) be T₁-space. To prove that {x} is MS-externally definable we must prove that {x }  ^c is MS-internally definable (minimal structure). Let y ∈ { x }  ^c   (y ≠ x) then there exists G,  H ∈ MS (U) such that y ∈ G,  x ∉ G and x ∈ H,  y ∉ H hence for all y ∈ { x }  ^c there exists G∈ MS (U) : y ∈ G ⊆ { x }. Since y is arbitrary, {x}  ^c  = ⋃ { G ∈ MS (U) , G ⊆ { x }  ^c  }. Then {x }  ^c is MS-open. Now, we prove the reverse implication, let {x} is MS-externally definable i.e, {x }  ^c is a MS-open set. Let x, y ∈ U such that x ≠ y. Then x ∈ {y}  ^c ,  y ∉ { y }  ^c and y ∈ { x }  ^c ,  x ∉ { x }  ^c since {x }  ^c and {y }  ^c are MS-open sets. Then (U, R, MS) is T₁-Space. □

Proposition 4.4. For a MSAS (U, R, MS) and preordering relation R. If (U, R, MS) is T₁- space, then A is MS-definable for all A ⊆ U.

Proof. Let A ⊂ U in order to prove A is MS-definable, we must prove that A is MS-internally (externally) definable set. Since A is MSAS and T₁- Space. Then {A} is MS-externally definable from Proposition 4.3. Also, A and U - A can be written as A = ⋃ _a∈A {a} and U - A = ⋃ _b∈U-A {b}. Then A and U - A are MS-externally definable sets from Proposition 3.8. Since U - A is MS-externally definable, A is MS-internally definable set. Thus A is MS-internally (externally) definable set. □

Proposition 4.5. An MSAST₁- Space with preordering relation is discrete approximation space.

Proof. Directly from 2 of Definition 4.1. □

Definition 4.6. For a MSAS (U, R, MS). The relation T₂ on U defined by xT₂y if and only if there exists G, H  ∈ MS (U) such that x ∈ G, y ∈ H and G∩ H = ∅ is called a separating relation.

Definition 4.7. For MSAS (U, R, MS) and T₂ is a separating relation on U. Then X ^{T ₂}  ={ y ∈ U : xT₂x, ∀ x ∈ X } is the separating set of A .

Definition 4.8. For MSAS (U, R, MS) and X, Y ⊆ U. Then XT₂X if and only if xT₂y for all x ∈ X, y ∈ Y.

Remark 4.9.

We notice that the separating relation T₂ is irreflexive relation and symmetric relation,

Proposition 4.10. For MSAS (U, R, MS) and X ⊆ U. Then

X ^{T ₂}  ⊆ X ^c ,

Proof.

y ∈ X ^{T ₂} if and only if yT₂X for all x ∈ X if and only if y ≠ x for all x ∈ X if and only if y ∈ X ^c  .

Proposition 4.11. For MSAS (U, R, MS) and X ⊆ U. Then

X ^{cT ₂}  ⊆ X,

Proof. Obvious by using Proposition 4.10. □

Proposition 4.12. For MSAS (U, R, MS) and X ⊆ U. Then X ^{T ₂} is minimal structure.

Proof. To prove that X ^{T ₂} is minimal structure, it is enough to prove that X ^{T ₂} can be written as union of minimal structure. Let y₁ ∈ X ^{T ₂} that is y₁T₂x for all x ∈ X Definition 4.7. there exists G₁, H₁∈ MS (U) ,  y₁ ∈ G₁ x ∈ H₁,  G₁ ∩ H₁ = ∅ Definition 4.6. But G₁∩ H₁ = ∅ for all x ∈ X such that x ∈ H₁ means G₁ ⊆ X ^{T ₂} . Then y₁ ∈ X ^{T ₂} implies there exists G₁ ∈ MS (U) : y₁ ∈ G₁ ⊆ X ^{T ₂} . Also, by similarity, y₂ ∈ X ^{T ₂} leads to exist G₂ ∈ MS (U) such that y₂ ∈ G₂ ⊆ X ^{T ₂} . Finally y _i  ∈ X ^{T ₂} implies there exists G _i  ∈ MS (U) such that y _i  ∈ G _i  ⊆ X ^{T ₂} . Hence ⋃ i = 1 n { y i } ⊆ ⋃ i = 1 n G i ⊆ X T 2 = ⋃ i = 1 n { y i } , i.e X T 2 = ⋃ i = 1 n { G i } . Then X ^{T ₂} is minimal structure. □

Example 4.13. Let (U, R, MS) be MSAS, U = {a, b, c, d}, where N (a) = {a, b}, N (b) = {b} , N (c) = {d} , N (d) = {a, c}. Then MS (U) = {∅ , {b} , {d} , {a, b} , {a, c} , U} and MS (U)  ^c  = {∅ , {b, d} , {c, d} , {a, b, c} , {a, c, d} , U}. Also, forme Table 4. we have LMS (X) = {∅ , {b} , {d} , {a, b} , {a, c} , {b, d} , {a, b, c} , {a, b, d} , {a, c, d} , U} UMS (X) = {∅ , {b} , {c} , {d} , {a, c} , {b, d} , {c, d} , {a, b, c} , {a, c, d} , U}

X ^{T ₂}  = {U, φ, {d} , {a, c} , {a, b}}

(X ^{T ₂} )  ^c  = {U, φ, {b, d} , {c, d} , {a, b . c}}

X ^{cT ₂}  = {U, φ, {b} , {d} , {a, c} , {a, b}}

Remark 4.14. Comparing lower operations LMS (X) and X ^{cT ₂} follows LMS (X) is the best lower approximation. Also, by comparing UMS (X) and (X ^{T ₂} )  ^c we note that UMS (X) is the best upper approximation.

Proposition 4.15. If (U, R, MS) be a MSAS, and X, Y ⊆ U. Then

X ∪ X ^{T ₂}  ⊆ U,

Proof.

We know that X ^{T ₂}  ⊆ X ^c . Then ⋃X ^{T ₂}  ⊆ X ∪ X ^c  = U .

Proposition 4.16. For a MSAS (U, R, MS) and preordering relation R. x T₂y if and only if N (A) ⋂ N (B) = ∅ .

Proof. Suppose that N (x)∩ N (y) ≠ ∅, and xT₂y. Then there exists E₁, E₂ ∈ Ω, x ∈ E₁, y ∈ E₂, E₁∩ E₂ = ∅. Since R is transitive, x ∈ E₁, y ∈ E₂ implies N (x) ⊆ E₁ and N (y) ⊆ E₂, but N (x)∩ N (y) ≠ ∅ implies E₁∩ E₂ ≠ ∅, leads to contradiction. Then N (x)∩ N (y) = ∅. The reverse implication follows immediately where N (x) , N (y) ∈ Ω and x ∈ N (x) , y ∈ N (y) by the reflexivity of R. □

Proposition 4.17. A MSAS (U, R, MS) is T₂-Space if and only if X ^{T ₂}  = X ^c , for all X ⊆ U.

Proof. By proposition 4.10, we have X ^{T ₂}  ⊆ X ^c , for all X ⊆ U. let y ∈ X ^c and y ≠ X, for all x ∈ Ximplies yhere exists G, H∈ MS (U) ,  x ∈ G, y ∈ H, G ∩ H = ∅. implies yT₂X, for all x ∈ X if and only if y ∈ X ^{T ₂} , X ^c  ⊆ X ^{T ₂} . Then X ^{T ₂}  = X ^c . □

Proposition 4.18. For MSAS (U, R, MS). Xis MS-definable if and only if (X ^{T ₂} )  ^c  = (X ^c )  ^{T ₂} .

Proof. First, let X ⊆ U is MS-definable and y ≠ X where x ∈ X and y ∈ X ^c . Since X and X ^c are minimal structure, xT₂y for all x ∈ X y ∈ X ^c . Then y ∈ X ^{T ₂} and x ∈ (X ^c )  ^{T ₂} . By Definition 4.7, we have X ^{T ₂}  = X ^c . Then (X ^{T ₂} )  ^c  = X = (X ^c )  ^{T ₂}  .

Second, let (X ^{T ₂} )  ^c  = X = (X ^c )  ^{T ₂} . By Proposition 4.12 and if (X ^c )  ^{T ₂}  = X. Then X is minimal structure. Also, if (X ^{T ₂} )  ^c  = X and X ^c is a minimal structure. Then X is MS-definable set. □

Proposition 4.19. For MSAS (U, R, MS) and X ⊆ U. The following statements are equivalent:

MSAS is T₂-Space,

Proof. 1 ⇒ 2 Since (U, R, MS) is T₂-Space, (X ^c )  ^{T ₂}  = X and X ^{T ₂}  = X ^c from Proposition 4.17 i.e., X ^{T ₂ c}  = X and X ^{cT ₂}  = X. Then X is MS-definable set from Proposition 4.18.

2 ⇒ 3: Obvious.

3 ⇒ 1: Let {x} and {y} are disjoint MS-definable set such that {x} ∩ { y } = ∅ and x∈ { x } and y∈ { y }, for all x, y ∈ U. Then (U, R, MS) is T₂-Space. □

Proposition 4.20. For MSAS (U, R, MS) and X ⊆ U. If R is an equivalence relation. Then X ^{T ₂}  = (UMS (X))  ^c .

Proof. Since R is an equivalence relation. If y ∈ X ^{T ₂} , then yT₂x for all x ∈ X and N (x)∩ N (y) = ∅. Also, if x ∉ N (y), for all x ∈ X and X∩ N (y) = ∅. Then y ∈ (UMS (X))  ^c . Therefore X ^{T ₂}  = (UMS (X))  ^c . □

Proposition 4.21. For MSAS (U, R, MS) and X ⊆ U and R an equivalence relation. If X is a rough set, then ((X ^c )  ^{T ₂} , (X ^{T ₂} )  ^c ) is the rough pair of X.

Proof. Since X ^{T ₂}  = (UMS (X))  ^c , for all X ∈ U. Then by Proposition 4.20 (X ^c )  ^{T ₂}  = (UMS (X ^c ))  ^c  = ((LMS (X))  ^c )  ^c  = LMS (X), and (X ^{T ₂} )  ^c  = UMS (X). But we know from Pawlak approximation space that (LMS (X) , UMS (A)) is the rough pair of a set A. Then ((X ^c )  ^{T ₂} , (X ^{T ₂} )  ^c ) is the rough pair of X. □

The approach presented in this paper is based on a generalized neighborhood system with a binary relation on a nonempty set by using minimal structure. The general relation used in this work is a generalization for all types of relations used in approximations. Moreover, the new approximation helps to get a new classification for definability. Also, we investigated the concepts of the separation axioms on MSAS which use separation properties in the information system as the process of approximation and definability. In future work, we use MSAS for the reduction and rule extraction in information systems.