Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subgraphs without coding or using assumption. The aim of this paper is to apply near concepts in the -closure approximation spaces. The basic notions of near approximations are introduced and sufficiently illustrated. Near approximations are considered as mathematical tools to modify the approximations of graphs. Moreover, proved results, examples, and counterexamples are provided.
1. Introduction
The theory of rough sets, proposed by Pawlak [1], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximation in rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules. The notions of closure operator and closure system are very useful tools in several sections of mathematics, as an example, in algebra [2β4], topology [5β7], and computer science theory [8, 9]. Many works have appeared recently, for example, in structural analysis [10, 11], in chemistry [12], and in physics [13]. The purpose of the present work is to put a starting point for the application of abstract topological graph theory in the rough set analysis. Also, we will integrate some ideas in terms of concept in topological graph theory. Topological graph theory is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics but also in many real-life applications. We believe that topological graph structure will be an important base for modification of knowledge extraction and processing.
2. Preliminaries
This section presents a review of some fundamental notions of Pawlakβs rough sets [1, 14, 15] and -closure spaces [10, 11].
2.1. Fundamental Notions of Uncertainty
Motivation for 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 , where is a set called the universe and is an equivalence relation [15, 16]. The equivalence classes of are also known as the granules, elementary sets or blocks; we will use to denote the equivalence class containing . In the approximation space, we consider two operators, the upper and lower approximations of subsets: let , then the lower approximation (resp., the upper approximation) of is given by
Boundary, positive, and negative regions are also defined:
In an approximation space , if and are two subsets of , then directly from the definitions of lower and upper approximations, we can get the following properties of the lower and upper approximations [15]:(1),
(2),
(3),
(4),
(5),
(6),
(7),
(8),
(9),
(10),
(11),
(12).
The inexactness of a set is due to the existence of a boundary region. The greater of the boundary region of a set, means the Pawlak [1], introduced the accuracy measure which is considered as a numerical characterization of imprecision. The following definition gives the accuracy measure of a subset in approximation space .
Definition 2.1. Let be an approximation space. The accuracy measure of a subset is defined by and define by
The accuracy measure is also called the accuracy of approximation.
2.2. Fundamental Notions of -Closure Spaces
In this section, we introduce the concepts of closure operators on digraphs; several known topological properties on the obtained -closure spaces are studied.
Definition 2.2 (see [10, 11]). Let be a digraph, its power set of all subgraphs of , and a mapping associating with each subgraph ; a subgraph is called the closure subgraph of such that
The operation is called graph closure operator, and the pair is called -closure space, where is the family of elements of . Evidently . The dual of the graph closure operator is the graph interior operator defined by for all subgraph . A family of elements of is called interior subgraph of and denoted by . It is clear that is a topological space. Evidently . Then the domain of is equal to the domain of and also . A subgraph of -closure space () is called closed subgraph if . It is called open subgraph if its complement is closed subgraph, that is, , or equivalently .
Example 2.3. Let be a digraph such that:,
, for more details (Table 1),
.
We obtain a new definition to construct topological closure spaces from -closure spaces by redefining graph closure operator on the resultant subgraphs as a domain of the graph closure operator and stop when the operator transfers each subgraph to itself.
Definition 2.4 (see [10, 11]). Let be a digraph and an operator such that:(a)It is called -closure operator if , -times, for every subgraph ,(b)it is called -topological closure operator if for all subgraph .The space () is called -closure space.
Example 2.5. Let be a digraph such that:,
for more details (see Table 2),
.
Proposition 2.6 (see [10]). Let () be a -closure space. If and are two subgraphs of such that , then
Proposition 2.7 (see [10]). Let () be a -closure space. If and are two subgraphs of , then(a),
(b).
Proposition 2.8 (see [10]). Let (, ) be a -closure space. If and are two subgraphs of , then(a), and(b).
Remark 2.9. The converse of Proposition 2.8 need not be true in general, as the following example (Exampleββ2.3 in [10]).
Definition 2.10 (see [10]). Let () be a -closure space and ; the boundary of is denoted by and is defined by
Proposition 2.11 (see [10]). Let () be a -closure space and , then(a),
(b),
(c),
(d).
By a similar way of definitions of regular open set [17], semiopen set [18], preopen set [19], Ξ³-open set [20] (b-open set [21]), Ξ±-open set [22], and Ξ²-open set [23] (=semi-pre-open set [24]), we introduce the following definitions which are essential for our present study. In -closure space (, ) the subgraph of is called(a)regular open subgraph [10] (briefly -osg) if ,(b) semiopen subgraph [10] (briefly -osg) if ,(c) preopen subgraph [10] (briefly -osg) if ,(d)Ξ³-open subgraph (briefly Ξ³-osg) if ,(e)Ξ±-open subgraph [10] (briefly Ξ±-osg) if ,(f)Ξ²-open subgraph [10] (briefly Ξ²-osg) if .
The complement of an -osg (resp., -osg, -osg, Ξ³-osg, Ξ±-osg, and Ξ²-osg) is called -closed subgraph (briefly -csg) (resp., S-csg, -csg, Ξ³-csg, Ξ±-csg, and Ξ²-csg).
The family of all -osgs (resp., -osgs, -osgs, Ξ³-osgs, Ξ±-osgs, and Ξ²-osgs) of (, ) is denoted by (resp., , , , , and ). All of , , , , and are larger than and closed under forming arbitrary union.
The family of all -csgs (resp., -csgs, -csgs, Ξ³-csgs, Ξ±-csgs, and Ξ²-csgs) of (, ) is denoted by () (resp., , , , , and ()).
The near closure (resp., near interior and near boundary) of a subgraph of in a -closure space is denoted by (resp. and ) and defined by
Proposition 2.12 (see [10]). Let () be -closure space, the implication and the families of near-open and near-closed graphs are given by following statements:(a),(b),(c),
(d).
3. Generalization of Pawlak Approximation Spaces
In this section we will generalize Pawlakβs concepts in the case of general relations. Hence, the approximation space with general relation on (i.e., closure operator on ) defines a uniquely -closure space (), where is the -closure space associated with . We will give this hypothesis in the following definition.
Definition 3.1. Let be an approximation space, where is a finite and nonempty universe graph, is a general relation on , and is the -closure space associated with . Then the triple is called a -closure approximation space. The following definition introduces the lower and the upper approximations in a -closure approximation space .
Definition 3.2. Let be a -closure approximation space and . The lower approximation (resp., the upper approximation) of is denoted by and is defined by
The following definition introduces new concepts of definability for a subgraph in a -closure approximation space .
Definition 3.3. Let be a -closure approximation space. If , then is called(a)totally -definable (-exact) graph if ,(b)internally -definable graph if ,(c)externally -definable graph if ,(d)-indefinable (-rough) graph if .
Proposition 3.4. Let be a -closure approximation space. If and are subgraphs of , then(1),(2),
(3),
(4),
(5),
(6),
(7),
(8),
(9),(10).
Proof. By using properties of -interior and -closure, the proof is obvious.
The following example illustrates that properties 11 and 12 which are introduced in Section 2.1 cannot be applied for this new generalization.
Example 3.5. Let be a -closure approximation space such that :, ,,
.
Let : , and : . Then
Thus,
Also,
Thus,
Lemma 3.6. Let () be a -closure space. Then
Proof. It follows from definition of -closure space.
Lemma 3.7. Let be a subgraph of in the -closure space (). Then if and only if for each subgraph and , then .
Proof. Let and for some . Assume . This implies that which is closed graph. Hence, , since and this leads to a contradiction. Therefore, . Suppose that for each and , . Let which is closed. Then there exists a closed graph uch that and . Hence, is open subgraph containing . Thus, and , that is, there exists a subgraph of such that , which leads to a contradiction. Therefore, .
Lemma 3.8. Let and be two subgraphs of in the -closure space (). If is open subgraph, then .
Proof. Let . If is open subgraph such that , then is an open subgraph and . Therefore, . Hence, the result.
Proposition 3.9. Let be a -closure approximation space. If and are subgraphs of , then(a),
(b).
Proof.
(a) We need to show that . Now,
Then,
Thus, by Lemma 3.6, we have
Therefore,
(b) We need to show that
Now,
Thus, by Lemma 3.6, we have
Hence, by Lemma 3.8, we have
Thus,
Therefore,
4. Near Lower and Near Upper in -Closure Approximation Spaces
In this section, we study approximation spaces from -closure view. We obtain some rules to find lower and upper approximations in several ways in approximation spaces with general relations. We will recall and introduce some definitions and propositions about some classes of near-open graphs which are essential for our present study.
Definition 4.1. Let be a -closure approximation space and . The near-lower approximation (-lower approximation) (resp., near-upper approximation (-upper approximation)) of is denoted by and is defined by
Proposition 4.2. Let be a -closure approximation space. If , then , for all .
Proof. The proofs of the five cases are similar, so we will only prove the case when . Now,
From (4.2) and (4.3) we get .
In general the above proposition is not true in the case of as the following example illustrates.
Example 4.3. Let be a -closure approximation space such that : ,,
.
Hence, . If : , then
Therefore,
Proposition 4.4. Let be a Gm- closure approximation space. If , then the implication between lower approximation and -lower approximation of are given by the following statement for all :(a),
(b).
Proof. By using Proposition 4.2, we get . We will prove . Now,
since . Thus,
Similarly we can prove the other cases.
Proposition 4.5. Let be a -closure approximation space. If , then the implication between upper approximation and -upper approximation of are given by the following statement for all ,(a),
(b).
Proof. By using Proposition 4.2, we get . We will prove . Now,
since . Thus,
Similarly we can prove the other cases.
Proposition 4.6. Let be a -closure approximation space. If and are two subgraphs of , then, for all ,(1),
(2),
(3),
(4),
(5),
(6),
(7),
(8),
(9).
Proof. By using properties of -interior and -closure for all , the proof is obvious.
In general, properties 3 and 4 which are introduced in Section 2.1 cannot be applied for -lower and -upper approximations, where . The following example illustrates this fact in the case of .
Example 4.7. Let be a -closure approximation space which is given in Example 2.3:,
.
If
then
but
Thus,
Also, if
then
but
Thus,
In general, properties 11 and 12 which are introduced in Section 2.1 cannot be applied for -lower and -upper approximations, where . The following example illustrates this fact in the case of .
Example 4.8. Let be a -closure approximation space which is given in Example 2.3. If
then
Thus,
Also,
Hence,
Lemma 4.9. Let be a -closure space. Then for all subgraph and .
Proof. It follows from definition near-open subgraphs in Gm-closure space.
Proposition 4.10. Let be a -closure approximation space. If and are subgraphs of , then
Proof. We need to show that
Now,
Then
Thus, by Lemma 4.9, we have
Therefore
In general, part (b) in Proposition 3.9 cannot be applied for -upper approximations for all . Example 4.11 (resp., Example 4.12) illustrates that part (b) in Proposition 3.9 cannot be applied in the case of .
Example 4.11. Let be a -closure approximation space which is given in Example 2.3. If
then
but
Hence,
Example 4.12. Let be a -closure approximation space which is given in Example 2.3:
If
then
but
Hence,
5. Near-Boundary Regions and Near Accuracy in -Closure Approximation Spaces
In this section we divide the boundary region into several levels. These levels help to decrease the boundary region. In the following definition we introduce the near boundary region of a subgraph of in a -closure approximation space .
Definition 5.1. Let be a -closure approximation space and . The near-boundary (-boundary) region of is denoted by and is defined by
Definition 5.2. Let be a -closure approximation space and . The near-positive (-positive) region of is denoted by and is defined by
Definition 5.3. Let be a -closure approximation space and . The near negative (briefly -negative) region of is denoted by and is defined by
Proposition 5.4. Let be a -closure approximation space. If , then
Proof. By using Proposition 4.2, the proof is obvious.
In general, the above proposition is not true in the case of as illustrated in the following example.
Example 5.5. Let be a -closure approximation space which is given in Example 2.3. If
then
Proposition 5.6. Let be a -closure approximation space. If , then the implication between boundary and -boundary of given by the following statement for all :(a),
(b).
Proof. By using Propositions 4.4 and 4.5, the proof is obvious.
Definition 5.7. Let be a -closure approximation space and a finite nonempty subgraph of . The near accuracy (-accuracy) of is denoted by and is defined by
Proposition 5.8. Let be a -closure approximation space. If is a finite nonempty subgraph of , then for all , where is the accuracy of .
Proof. By using Proposition 4.2, the proof is obvious.
In general, the above proposition is not true in the case of . This fact is illustrated in the following example.
Example 5.9. Let be a -closure approximation space which is given in Example 4.3. If
then
Thus,
Proposition 5.10. Let be a -closure approximation space. If , then the implication between accuracy and -accuracy of is given by the following statement for all :(a),
(b).
Proof. By using Propositions 4.4 and 4.5, the proof is obvious.
6. Rough and Near-Rough Cluster Vertices in -Closure Approximation Spaces
In this section, we introduce the definitions of definability of graphs, rough cluster vertices and near-rough cluster vertices in approximation spaces with general relations. The following definition introduces new concepts of definability for a subgraph in a -closure approximation space .
Definition 6.1. Let be a -closure approximation space. If , then is called(a)totally -definable (-exact) graph if ,(b)internally -definable graph if ,(c)externally -definable graph if ,(d)-indefinable (-rough) graph if , where .
Example 6.2. Let be a -closure approximation space such that , ,
.
Let : , , then, for , we get
Thus, is an -indefinable (-rough) graph and -definable (-exact) graph. The following definition introduces the concept of rough cluster vertices of a subgraph of in a -closure approximation .
Definition 6.3. Let be a -closure approximation space. The vertex is said to be a rough cluster vertex of a subgraph of if, for all subgraph of such that , . The graph of all rough cluster vertices of is denoted by and is called the rough derived graph of .
Theorem 6.4. Let be a -closure approximation space. Then a subgraph of is closed if and only if .
Proof. Suppose that is a closed subgraph of , and let (i.e., ). Then is open subgraph. Thus, and . Hence, . Therefore, . Let . To show that is a closed subgraph of , let . Then , and hence there exists a subgraph such that and . But , hence . So and . Thus, is a union of open graphs, which is open. Hence, is closed subgraph of .
Example 6.5. Let be a -closure approximation space such that : , , ,
.
If : , , then . Thus, and is closed subgraph of . The following definition introduces the concept of near-rough (-rough) cluster vertices of a subgraph of in a -closure approximation space for all .
Definition 6.6. Let be a -closure approximation space. The vertex is said to be near-rough (-rough) cluster vertex of a subgraph of for all , if, for all subgraph of such that , . The graph of all -rough cluster vertices of is denoted by and is called the -rough derived graph of for all .
Theorem 6.7. Let be a -closure approximation space. Then a subgraph of is a -closed for all if and only if .
Proof. The proofs of the five cases are similar, so we will only prove the case when . Suppose that is a -closed subgraph of , and let (i.e., ). Then . Thus, and . Hence, . Therefore, . Let . To show that is a -closed subgraph of , let , then , and hence there exists a subgraph such that and . But , hence . So and . Thus, is a union of -open graphs, which is -open. Hence, is -closed subgraph of .
Example 6.8. Let be a -closure approximation space which is given in Example 6.5. If . Then thus and is -closed subgraph of . In general, Theorem 6.7 cannot be satisfied in the case of , as the following example illustrates.
Example 6.9. Let be a -closure approximation space which is given in Example 6.5. If . Then , thus . But is not an -closed subgraph of , since .
Theorem 6.10. Let be a subgraph of in the -closure approximation space . Then if and only if, for each and , .
Proof. () Let and for some . Assume . This implies that . But which is closed graph. Hence, , since and this leads to a contradiction. Therefore, . Suppose that, for each and , . Let . But which is closed. Then there exists a closed graph such that and . Hence, is open graph containing . Thus,
that is, there exists a subgraph such that , which leads to a contradiction. Therefore, .
Theorem 6.11. Let be a subgraph of in the -closure approximation space . Then for all if and only if, for each and , .
Proof. The proof is similar to the proof of Theorem 6.10.
Theorem 6.12. Let be a subgraph of in the -closure approximation space . Then .
Proof. By Theorem 6.4, we get . Then
For the converse inclusion, let , then either and hence or . Hence, by Theorem 6.10 for each , , we get . Then and hence . Thus, . Therefore, .
Theorem 6.13. Let be a subgraph of in the -closure approximation space . Then for all .
Proof. The proof is similar to the proof of Theorem 6.12.
7. Conclusions
In this paper, we used -topological concepts to introduce a generalization of Pawlak approximation space. Concepts of definability for subgraphs in -approximation spaces are introduced. Several types of approximations which are called near approximations are mathematical tools to modify the approximations. The suggested methods of near approximations open way for constructing new types of lower and upper approximations.
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,
.
,
.
,
.
Let : , and : . Then
Thus,
Also,
Thus,
,
.
Hence, . If : , then
Therefore,
,
.
Let : , , then, for , we get
Thus, is an -indefinable (-rough) graph and -definable (-exact) graph.
,
.
If : , , then . Thus, and is closed subgraph of . 