On an Open Problem for Definable Subsets of Covering Approximation Spaces
Let (U;D) be a Gr-covering approximation space
(U; C) with covering lower approximation operator D and covering
upper approximation operator D. For a subset X of U, this paper
investigates the following three conditions: (1) X is a definable subset
of (U;D); (2) X is an inner definable subset of (U;D); (3) X is an
outer definable subset of (U;D). It is proved that if one of the above
three conditions holds, then the others hold. These results give a
positive answer of an open problem for definable subsets of covering
approximation spaces.
[1] Z. Bonikowski, E. Bryniarski and U. Wybraniec, Extensions and intentions
in the rough set theory, Information Sciences, 107(1998), 149-167.
[2] X. Ge, Further investigations on higher mathematics scores for Chinese
university students, International Journal of Computational and Mathematical
Sciences, 3(2009), 181-185.
[3] X. Ge, An application of covering approximation spaces on network
security, Computers and Mathematics with Applications, 60(2010), 1191-
1199.
[4] X. Ge, J. Li and Y. Ge, Some separations in covering approximation, International
Journal of Computational and Mathematical Sciences, 4(2010),
156-160.
[5] X. Ge and Z. Li, Definable subsets in covering approximation spaces,
International Journal of Computational and Mathematical Sciences,
5(2011), 31-34.
[6] X. Ge and J. Qian, Some investigations on higher mathematics scores
for Chinese university students, International Journal of Computer and
Information Engineering, 3(2009), 209-212.
[7] E. Lashin, A. Kozae, A. Khadra and T. Medhat, Rough set theory
for topological spaces, International Journal of Approximate Reasoning,
40(2005), 35-43.
[8] Z. Pawlak, Rough sets, International journal of computer and information
science, 11(1982), 341-356.
[9] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data,
Kluwer Academic Publishers, 1991.
[10] Z. Pawlak, Rough classification, International journal of Humancomputer
Studies, 51(1999), 369-383.
[11] Z. Pawlak and A. Skowron, Rudiments of rough sets, Information
Sciences, 177(2007), 3-27.
[12] Z. Pawlak and A. Skowron, Rough sets: Some extensions, Information
Sciences, 177(2007), 28-40.
[13] Z. Pawlak and A. Skowron, Rough sets and Boolean reasoning, Information
Sciences, 177(2007), 41-73.
[14] D. Pei, On definable concepts of rough set models, Information Sciences,
177(2007), 4230-4239.
[15] J. A. Pomykala, Approximation operations in approximation spaces,
Bull. Pol. Acad. Sci., 35(1987), 653-662.
[16] K. Qin, Y. Gao and Z. Pei, On covering rough sets, Lecture Notes in
Artificial Intelligence, 4481(2007), 34-41.
[17] P. Samanta and M. Chakraborty, Covering based approaches to rough
sets and implication lattices, Lecture Notes in Artificial Intelligence,
5908(2009), 127-134.
[18] Y. Yao, Views of the theory of rough sets in finite universes, International
Journal of Approximate Reasoning, 15(1996), 291-317.
[19] Y. Yao, Relational interpretations of neighborhood operators and rough
set approximation operators, Information Sciences, 111(1998), 239-259.
[20] Y. Yao, On generalizing rough set theory, Lecture Notes in AI,
2639(2003), 44-51.
[21] Z. Yun, Xun Ge and X. Bai, Axiomatization and conditions for neighborhoods
in a covering to form a partition, Information Sciences, (2011),
doi:10.1016/j.ins.2011.01.013.
[22] W. Zakowski, Approximations in the Space (u, Π), Demonstratio Mathematica,
16(1983), 761-769.
[23] W. Zhu, Topological approaches to covering rough sets, Information
Sciences, 177(2007), 1499-1508.
[24] W. Zhu, Relationship between generalized rough sets based on binary
relation and covering, Information Sciences, 179(2009), 210-225.
[25] W. Zhu and F. Wang, Reduction and axiomatization of covering generalized
rough sets, Information Sciences, 152(2003), 217-230.
[26] W. Zhu and F. Wang, On Three Types of Covering Rough Sets, IEEE
Transactions on Knowledge and Data Engineering, 19(2007), 1131-1144.
[1] Z. Bonikowski, E. Bryniarski and U. Wybraniec, Extensions and intentions
in the rough set theory, Information Sciences, 107(1998), 149-167.
[2] X. Ge, Further investigations on higher mathematics scores for Chinese
university students, International Journal of Computational and Mathematical
Sciences, 3(2009), 181-185.
[3] X. Ge, An application of covering approximation spaces on network
security, Computers and Mathematics with Applications, 60(2010), 1191-
1199.
[4] X. Ge, J. Li and Y. Ge, Some separations in covering approximation, International
Journal of Computational and Mathematical Sciences, 4(2010),
156-160.
[5] X. Ge and Z. Li, Definable subsets in covering approximation spaces,
International Journal of Computational and Mathematical Sciences,
5(2011), 31-34.
[6] X. Ge and J. Qian, Some investigations on higher mathematics scores
for Chinese university students, International Journal of Computer and
Information Engineering, 3(2009), 209-212.
[7] E. Lashin, A. Kozae, A. Khadra and T. Medhat, Rough set theory
for topological spaces, International Journal of Approximate Reasoning,
40(2005), 35-43.
[8] Z. Pawlak, Rough sets, International journal of computer and information
science, 11(1982), 341-356.
[9] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data,
Kluwer Academic Publishers, 1991.
[10] Z. Pawlak, Rough classification, International journal of Humancomputer
Studies, 51(1999), 369-383.
[11] Z. Pawlak and A. Skowron, Rudiments of rough sets, Information
Sciences, 177(2007), 3-27.
[12] Z. Pawlak and A. Skowron, Rough sets: Some extensions, Information
Sciences, 177(2007), 28-40.
[13] Z. Pawlak and A. Skowron, Rough sets and Boolean reasoning, Information
Sciences, 177(2007), 41-73.
[14] D. Pei, On definable concepts of rough set models, Information Sciences,
177(2007), 4230-4239.
[15] J. A. Pomykala, Approximation operations in approximation spaces,
Bull. Pol. Acad. Sci., 35(1987), 653-662.
[16] K. Qin, Y. Gao and Z. Pei, On covering rough sets, Lecture Notes in
Artificial Intelligence, 4481(2007), 34-41.
[17] P. Samanta and M. Chakraborty, Covering based approaches to rough
sets and implication lattices, Lecture Notes in Artificial Intelligence,
5908(2009), 127-134.
[18] Y. Yao, Views of the theory of rough sets in finite universes, International
Journal of Approximate Reasoning, 15(1996), 291-317.
[19] Y. Yao, Relational interpretations of neighborhood operators and rough
set approximation operators, Information Sciences, 111(1998), 239-259.
[20] Y. Yao, On generalizing rough set theory, Lecture Notes in AI,
2639(2003), 44-51.
[21] Z. Yun, Xun Ge and X. Bai, Axiomatization and conditions for neighborhoods
in a covering to form a partition, Information Sciences, (2011),
doi:10.1016/j.ins.2011.01.013.
[22] W. Zakowski, Approximations in the Space (u, Π), Demonstratio Mathematica,
16(1983), 761-769.
[23] W. Zhu, Topological approaches to covering rough sets, Information
Sciences, 177(2007), 1499-1508.
[24] W. Zhu, Relationship between generalized rough sets based on binary
relation and covering, Information Sciences, 179(2009), 210-225.
[25] W. Zhu and F. Wang, Reduction and axiomatization of covering generalized
rough sets, Information Sciences, 152(2003), 217-230.
[26] W. Zhu and F. Wang, On Three Types of Covering Rough Sets, IEEE
Transactions on Knowledge and Data Engineering, 19(2007), 1131-1144.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56124", author = "Mei He and Ying Ge and Jingyu Qian", title = "On an Open Problem for Definable Subsets of Covering Approximation Spaces", abstract = "Let (U;D) be a Gr-covering approximation space
(U; C) with covering lower approximation operator D and covering
upper approximation operator D. For a subset X of U, this paper
investigates the following three conditions: (1) X is a definable subset
of (U;D); (2) X is an inner definable subset of (U;D); (3) X is an
outer definable subset of (U;D). It is proved that if one of the above
three conditions holds, then the others hold. These results give a
positive answer of an open problem for definable subsets of covering
approximation spaces.", keywords = "Covering approximation space, covering approximation operator, definable subset, inner definable subset, outer definable subset.", volume = "5", number = "3", pages = "360-3", }
