Applied Mathematics
Vol.07 No.14(2016), Article ID:70078,9 pages
10.4236/am.2016.714136
γ and β Approximations via General Ordered Topological Spaces
Mohamed Abo-Elhamayel
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 9 May 2016; accepted 22 August 2016; published 25 August 2016
ABSTRACT
In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.
Keywords:
Rough Sets, Approximations, Ordered Topological Spaces
1. Introduction
Rough set theory was first proposed by Pawlak for dealing with vagueness and granularity in information systems. Various generalizations of Pawlak s rough set have been made by replacing equivalence relations with kinds of binary relations and many results about generalized rough set with the universe being finite were obtained [1] - [7] . An interesting and natural research topic in rough set theory is studying it via topology [8] [9] . Neighborhood systems were first applied in generalizing rough sets in 1998 by T. Y. Lin as a generalization of topological connections with rough sets. Lin also introduced the concept of granular computing as a form of topological generalizations [10] - [13] . In this paper, we give the concept of g, b via topological ordered spaces and studied their properties which may be viewed as a generalization of previous studies in general approximation spaces, as if we take the partially ordered relation as an equal relation, we obtain the concepts in general approximation spaces [14] .
2. Preliminaries
In this section, we give an account of the basic definitions and preliminaries to be used in the paper.
Definition 2.1 [15] . A subset A of U, where is a partially ordered set is said to be increasing (resp. decreasing) if for all
and
such that
(resp.
) imply
.
Definition 2.2 [15] . A triple is said to be a topological ordered space, where
is a topological space and
is a partial order relation on U.
Definition 2.3 [16] . Information system is a pair where U is a non-empty finite set of objects and
is a non-empty finite set of attributes.
Definition 2.4 [17] . A non-empty set U equipped with a general relation which generates a topology
on U and a partially order relation
written as
is said to be general ordered topological approximation space (for short, GOTAS).
Definition 2.5 [18] . Let be a GOTAS and
. We define:
(1),
is the greatest increasing open subset of A.
(2),
is the greatest decreasing open subset of A.
(3),
is the smallest increasing closed superset of A.
(4),
is the smallest decreasing closed superset of A.
(5) (resp.
) and
)resp.
) is R-increasing (resp. decreasing) accuracy.
Definition 2.6 [17] . Let be a GOTAS and
. We define:
(1),
is called R-increasing semi lower.
(2),
is called R- increasing semi upper.
(3),
is called R-decreasing semi lower.
(4),
is called R-decreasing semi upper.
A is R- increasing (resp. decreasing) semi exact if (resp.
), otherwise A is R- increasing (resp. decreasing) semi rough.
Proposition 2.7 [18] . Let be a GOTAS and
. Then
(1).
(2).
3. New Approximations and Their Properties
In this section, we introduce some definitions and propositions about near approximations, near boundary regions via GOTAS which is essential for a present study.
Definition 3.1. Let be a GOTAS and
. We define:
(1),
is called R-increasing
lower.
(2),
is called R-increasing
upper.
(3),
is called R-decreasing
lower.
(4),
is called R-decreasing
upper.
A is R-increasing (resp. R-decreasing) exact if
(resp.
) otherwise A is R-increasing (resp. R-decreasing)
rough.
Proposition 3.2. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
(3) (
).
Proof.
(1) Omitted.
One can prove the case between parentheses.
Proposition 3.3. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
(3) (
).
Proof.
(1) Easy.
One can prove the case between parentheses.
Proposition 3.4. Let be a GOTAS and
. If A is R-increasing (resp. decreasing) exact then A is R-increasing (resp. decreasing)
exact.
Proof.
Let A be R-increasing exact. Then, thus
and
. Therefore
.
One can prove the case between parentheses.
R-increasing (resp. decreasing) exact R-increasing (resp. decreasing)
exact.
Proposition 3.5. Let be a GOTAS and
. Then
.
Proof.
Since and
, then
. There-
fore,. Thus
.
One can prove the case between parentheses.
Proposition 3.6. Let be a GOTAS and
. Then
.
Proof. Since and
, then
. Thus
.
Therefore. Hence
.
Proposition 3.7. Let be a GOTAS and
. Then
.
Proof. Let. Then
and
. Therefore
and
.
Thus. Hence
.
One can prove the case between parentheses.
Proposition 3.8. Let be a GOTAS and
. Then
.
Proof.
Let. Then
and
. Therefore
and
or
.
Thus. Hence
.
One can prove the case between parentheses.
Proposition 3.9. Let be a GOTAS and
. Then
.
Proof.
Let. Then
and
. Therefore
.
Thus.
Proposition 3.10. Let be a GOTAS and
. Then
.
Proof. Omitted.
Definition 3.11. Let be a GOTAS and
. We define:
(1),
is called R-increasing
lower.
(2),
is called R-increasing
upper.
(3),
is called R-decreasing
lower.
(4),
is called R-decreasing
upper.
A is R-increasing (decreasing) exact if
(resp.
), otherwise A is R-increasing (decreasing)
rough.
Proposition 3.12. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
(3) (
).
Proof.
(1) Omitted.
One can prove the case between parentheses.
Proposition 3.13. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
(3) (
).
Proof.
(1) Easy.
One can prove the case between parentheses.
Proposition 3.14. Let be a GOTAS and
. If A is R-increasing (resp. decreasing) exact then A is b-increasing (resp. decreasing) exact.
Proof.
Let A be R-increasing exact. Then. Therefore
,
. Thus
. Hence A is R-increasing
exact.
One can prove the case between parentheses.
Proposition 3.15. Let be a GOTAS and
. Then
.
Proof.
Since and
. Then
.
Therefore. Thus
.
One can prove the case between parentheses.
Proposition 3.16. Let be a GOTAS and
. Then
.
Proof. Since and
. Then
. Thus
.
Therefore. Hence
.
Definition 3.17. Let be a GOTAS and
. Then
(1) (resp.
), is increasing (resp. decreasing) j boundary region.
(2) (resp.
), is increasing (resp. decreasing) j positive region.
(3) ( resp.
), is increasing (resp. decreasing) j negative region. Where
the near lower approximations s.t.
.
Proposition 3.18. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
Proof.
One can prove the case between parentheses.
Proposition 3.19. Let be a GOTAS and
. Then
(1) (
).
(2) (
).
Proof.
One can prove the case between parentheses.
Proposition 3.20. Let be a GOTAS and
. Then
.
Proof.
Let. Then
. Therefore
. Thus
and thus.
Hence
(1).
Since, then
. Therefore
.
Thus, and thus
. Hence
(2)
From (1) and (2) we have,
.
One can prove the case between parentheses.
Proposition 3.21. Let be a GOTAS and
. Then
.
Proof.
Let. Then
. Therefore
or
. Thus
or
. So
, and so
.
Thus. Hence
. (1)
Since,
or
, then
. Therefore
(2)
From (1) and (2) we have,.
One can prove the case between parentheses.
Definition 3.22. Let be a GOTAS and A is a non-empty finite subset of U. Then the increasing (decreasing) j accuracy of a finite non-empty subset A of U is given by:
,
.
Proposition 3.23. Let be a GOTAS and
non-empty finite subset of
. Then we have
, for all
, where
Proof. Omitted.
In the following example we illustrate most of the properties that have been proved in the previous propositions.
Example 3.24. Let,
,
,
and
For, we have:
,
,
,
.
.
.
,
,
Proposition 3.25. Let be a GOTAS and
. Then we have
Proof. Omitted.
Remark 3.26..
Remark 3.27..
Proposition 3.28. Let be a GOTAS and
be a non-empty finite subset of
. Then
(
).
Proof. Omitted.
Proposition 3.28. Let be a GOTAS and
. Then
Proof. Let. Then
and
.
Therefore and [
or
]. Thus
and
and
thus and
. Hence
. Therefore
.
One can prove the case between parentheses.
4. Conclusion
In this paper, we generalize rough set theory in the framework of topological spaces. Our results in this paper became the results about of,
approximation in [2] in the case of
is the equal relation. Also, the new approximation which we give became as Pawlak s approximation in the case of
is the equal relation and R is the equivalence relation. This theory brings in all these techniques to information analysis and knowledge processing.
Cite this paper
Mohamed Abo-Elhamayel, (2016) γ and β Approximations via General Ordered Topological Spaces. Applied Mathematics,07,1580-1588. doi: 10.4236/am.2016.714136
References
- 1. Abo-Elhamayel, M. (2013) Near Approximations via General Ordered Topological Spaces. Journal of Advances in Mathematics, 2, 61-72.
- 2. El-Shafei, M.E., Kozae, A.M. and Abo-Elhamayel, M. (2013) Rough Set Approximations via Topological Ordered Spaces. Annals of Fuzzy Sets, Fuzzy Logic and Fuzzy Systems, 2, 49-60.
- 3. Pawlak, Z. (1991) Rough Sets, Theoretical Aspects of Reasoning about Data. Kluwer Academic, Boston.
- 4. Nachbin, L. (1965) Topology and or$der van Nostrand Mathematical s$tudies. Princeton, New Jersey.
- 5. AbdEl-Monsef, M.E., et al. (2009) On Near Open Sets and Near Approximations. Journal of institute of Mathematics and Computer Sciences, 20, 99-109.
- 6. Lin, T.Y. (2012) Uncertainty and Knowledge Theories New Era in Granular Computing. IEEE International Conference on Granular Computing, Hangzhou, 11-13 Aug. 2012, 8-11.
http://dx.doi.org/10.1109/grc.2012.6468707 - 7. Lin, T.Y. (1999) Granular Computing: Fuzzy Logic and Rough Sets, Computing with Words in Information Intelligent Systems. Physica-Verlag, Heidelberg, 183-200.
- 8. Lin, T.Y. (1998) Granular Computing on Binary Relations II: Rough Set Representations and Belief Functions. Rough Sets in Knowledge Discovery, PhysicaVerlag.
- 9. Lin, T.Y. (1998) Granular Computing on Binary Relations I: Data Mining and Neighborhood Systems. Rough Sets In Knowledge Discovery, Physica-Verlag, Heidelberg, 107-121.
- 10. Polkowski, L. (2002) Rough Sets: Mathematical Foundations. Physica-Verlag, Heidelberg.
http://dx.doi.org/10.1007/978-3-7908-1776-8 - 11. Lashin, E.F., Kozae, A.M., Abokhadra, A.A. and Medhat, T. (2005) Rough Set Theory for Topological Spaces. International Journal of Approximate Reasoning, 40, 35-43.
http://dx.doi.org/10.1016/j.ijar.2004.11.007 - 12. Zhu, W. (2007) Generalized Rough Sets Based on Relations. Information Sciences, 177, 4997-5011.
http://dx.doi.org/10.1016/j.ins.2007.05.037 - 13. Zhang, W.X., Wu, W.Z., Liang, J.Y. and Li, D.Y. (2001) Rough Sets Theory and Methods. Science Press, Beijing.
- 14. Yao, Y.Y. (1998) Constructive and Algebraic Methods of the Theory of Rough Sets. Information Sciences, 109, 21-47.
http://dx.doi.org/10.1016/S0020-0255(98)00012-7 - 15. Pei, D.W. (2007) On Definable Concepts of Rough Set Models. Information Sciences, 177, 4230-4239.
http://dx.doi.org/10.1016/j.ins.2007.01.020 - 16. Pawlak, Z. and Skowron, A. (2007) Rough Sets: Some Extensions. Information Sciences, 177, 28-40.
http://dx.doi.org/10.1016/j.ins.2006.06.006 - 17. Pawlak, Z. and Skowron, A. (2007) Rudiments of Rough Sets. Information Sciences, 177, 3-27.
http://dx.doi.org/10.1016/j.ins.2006.06.003 - 18. El-Shafei, M.E., Kozae, A.M. and Abo-Elhamayel, M. (2013) Semi Ordered Topological Approximations of Rough Sets. Annals of Fuzzy Sets, Fuzzy Logic and Fuzzy Systems, 2, 61-72.