Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets

Affiliation(s)

Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

ABSTRACT

Functions are a means to link or transport from a world to another world may be similarly or completely different from the other world. In this paper we addressed the issue of rough functions and the possibility of transfer it from the real line to the topological abstract view that can be applied to intelligent information systems. The rough function approach has not been studied much specially from a topological point of view. Here we developed a new type of topological generalizations of rough functions with reference to how it is used in medical applications. Considering that the function is in the original a relation can be based on a review of all circular functions from the perspective of relations. Accordingly, the dream that the generalizations of rough functions are transferred to all papers prior to a comprehensive computer application.

Functions are a means to link or transport from a world to another world may be similarly or completely different from the other world. In this paper we addressed the issue of rough functions and the possibility of transfer it from the real line to the topological abstract view that can be applied to intelligent information systems. The rough function approach has not been studied much specially from a topological point of view. Here we developed a new type of topological generalizations of rough functions with reference to how it is used in medical applications. Considering that the function is in the original a relation can be based on a review of all circular functions from the perspective of relations. Accordingly, the dream that the generalizations of rough functions are transferred to all papers prior to a comprehensive computer application.

Cite this paper

A. Salama and H. Abu-Donia, "Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets,"*Journal of Intelligent Learning Systems and Applications*, Vol. 4 No. 2, 2012, pp. 127-134. doi: 10.4236/jilsa.2012.42012.

A. Salama and H. Abu-Donia, "Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets,"

References

[1] Z. Pawlak, “Rough Sets,” International Journal of Parallel Programming, Vol. 11, No. 5, 1982, pp. 341-356. doi:10.1007/BF01001956

[2] X. Wang, E. C. C. Tsang, S. Zhao, D. Chen and S. Yeung, “Learning Fuzzy Rules from Fuzzy Samples Based on Rough Set Technique,” Information Sciences, Vol. 177, No. 15, 2007, pp. 4493-4514. doi:10.1016/j.ins.2007.04.010

[3] S. Zhao and E. C. C. Tsang, “On Fuzzy Approximation Operators in Attribute Reduction with Fuzzy Rough Sets,” Information Sciences, Vol. 178, No. 16, 2008, pp. 3163-3176.

[4] L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X

[5] Z. Bonikowski, “Algebraic Structures of Rough Sets,” In: W. Ziarko, Ed., Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer, London, 1994, pp. 243-247. doi:10.1007/978-1-4471-3238-7_29

[6] E. Bryniaski, “A Calculus of Rough Sets of The First Order,” Bulletin of the Polish Academy of Sciences, Vol. 16, 1989, pp. 71-77.

[7] Y. Y. Yao, “Constructive and Algebraic Methods of Theory of Rough Sets,” Information Sciences, Vol. 109, No. 1-4, 1998, pp. 21-47. doi:10.1016/S0020-0255(98)00012-7

[8] A. Skowron and J. Stepaniuk, “Tolerance Approximation Spaces,” Fund Information, Vol. 27, No. 2-3, 1996, pp. 245-253.

[9] E. F. Lashin, A. M. Kozae, A. A. Abo Khadra and T. Medhat, “Rough Set Theory for Topological Spaces,” International Journal of Approximate Reasoning, Vol. 40, No. 1-2, 2005, pp. 35-43. doi:10.1016/j.ijar.2004.11.007

[10] K. Qin and Z. Pei, “On the topological Properties of Fuzzy Rough Sets,” Fuzzy Sets and Systems, Vol. 151, No. 3, 2005, pp. 601-613. doi:10.1016/j.fss.2004.08.017

[11] A. Wasilewska, “Topological Rough Algebras,” In: T. Y. Lin and N. Cercone, Eds., Rough Sets and Data Mining, Kluwer Academic Publishers, Boston, 1997, pp. 425-441. doi:10.1007/978-1-4613-1461-5_21

[12] W. Zhu, “Topological Approaches to Covering Rough Sets,” Information Sciences, Vol. 177, No. 15, 2007, pp. 1499-1508. doi:10.1016/j.ins.2006.06.009

[13] T. J. Li, Y. Leung and W. X. Zhang, “Generalized Fuzzy Rough Approximation Operators Based on Fuzzy Coverings,” International Journal of Approximate Reasoning, Vol. 48, No. 3, 2008, pp. 836-856. doi:10.1016/j.ijar.2008.01.006

[14] R. Biswas, “On Rough Sets and Fuzzy Rough Sets,” Bulletin of the Polish Academy of Sciences, Vol. 42, 1992, pp. 343-349.

[15] G. Liu, “Generalized Rough Sets over Fuzzy Lattices,” Information Sciences, Vol. 178, No. 6, 2008, pp. 16511662. doi:10.1016/j.ins.2007.11.010

[16] A. M. Rolka and L. Rolka, “Fuzzy Rough Approximations of Process Data,” International Journal of Approximate Reasoning, Vol. 49, No. 2, 2008, pp. 301-315. doi:10.1016/j.ijar.2007.03.016

[17] M. Banerjee and S. K. Pal, “Roughness of a Fuzzy Set,” Information Sciences, Vol. 93, No. 3-4, 1995, pp. 235246. doi:10.1016/0020-0255(96)00081-3

[18] R. Biswas, “On Rough Fuzzy Sets,” Bulletin of the Polish Academy of Sciences, Vol. 42, 1994, pp. 352-355.

[19] D. Dubois and H. Prade, “Rough Fuzzy Sets and Fuzzy Rough Sets,” International Journal of General Systems, Vol. 17, No. 2-3, 1990, pp. 191-208. doi:10.1080/03081079008935107

[20] C. Degang, Y. Wenxia and Li Fachao, “Measures of General Fuzzy Rough Sets on a Probabilistic Space,” Information Sciences, Vol. 178, No. 16, 2008, pp. 3177-3187. doi:10.1016/j.ins.2008.03.020

[21] Z. Gong, B. Sun and D. Chen, “Rough Set Theory for the Interval-Valued Fuzzy Information Systems,” Information Sciences, Vol. 178, No. 8, 2008, pp. 1968-1985. doi:10.1016/j.ins.2007.12.005

[22] Y. Yang and C. Hinde, “A New Extension of Fuzzy Sets Using Rough Sets: R-Fuzzy Sets,” Information Sciences, Vol. 180, No. 3, 2010, pp. 354-365. doi:10.1016/j.ins.2009.10.004

[23] P. Bhattacharya and B. K. Lahiri, “Semi Generalized Closed Sets in Topology,” Indian Journal of Mathematics, Vol. 29, 1987, pp. 373-382.

[24] A. Skowron, “On Topology Information Systems,” Bulletin of the Polish Academy of Sciences, Vol. 3, 1989, pp. 87-90.

[25] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publishers, Boston, 1991.

[26] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, “On Pre-Continuous and Weak Pre Continuous Mappings,” Mathematical Physical and Engineering Sciences, Vol. 53, 1982, pp. 47-53.

[27] O. Najsted, “On Some Classes of Nearly Open Sets,” Pacific Journal of Mathematics, Vol. 15, 1965, pp. 961970.

[28] J. R. Munkres, “Topology, a First Course,” Prentice-Hall, Upper Saddle River, 1975.

[29] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, “?-Open Sets and ?-Continuous Mappings,” Bulletin of the Faculty of Science, Assiut University, Asyut, 1983.

[30] D. Andrijevic, “On b-Open Sets,” Matematicki Vesnik, Vol. 48, 1996, pp. 59-64.

[31] A. S. Mashhour, I. A. Hasanein and S. N. El Deeb, “A Note on α-Continuous and α-Open Mappings,” Acta Mathematica Hungarica, Vol. 41, No. 3-4, 1983, pp. 213218.

[32] J. Kortelainen, “On the Relationship between Modified Sets, Topological Spaces and Rough Sets,” Fuzzy Sets and Systems, Vol. 61, No. 1, 1994, pp. 91-95. doi:10.1016/0165-0114(94)90288-7

[33] Y. Y. Yao, “Two Views of the Theory of Rough Sets in Finite Universes,” International Journal of Approximate Reasoning, Vol. 15, No. 4, 1996, pp. 291-317. doi:10.1016/S0888-613X(96)00071-0

[34] Y. Y. Yao and T. Y. Lin, “Generalization of Rough Sets Using Modal Logics,” Intelligent Automation & Soft Computing, Vol. 2, 1996, pp. 103-120.

[35] Z. Pawlak, “On Rough Relations,” Bulletin of the Polish Academy of Sciences, Vol. 34, 1986, pp. 9-10.

[36] Z. Pawlak, “Rough Sets, Rough Relations and Rough Functions,” Bull of the Polish Academy of Sciences, Vol. 13, 1996, pp. 15-19.

[37] A. S. Salama, “Bitopological Rough Approximations with Medical Applications,” Journal of King Saud University (Science), Vol. 22, No. 3, 2010, pp. 177-183. doi:10.1016/j.jksus.2010.04.010

[38] J. Kelley, “General Topology,” Van Nostrand Company, New York, 1955.

[39] N. Levine, “Semi Open Sets and Semi Continuous Mappings in Topological Spaces,” American Mathematical Monthly, Vol. 70, No. 1, 1963, pp. 36-41. doi:10.2307/2312781

[40] M. H. F. Zarandi, et al., “A Fuzzy Expert System Architecture for Intelligent Tutoring Systems: A Cognitive Mapping Approach,” Journal of Intelligent Learning Systems and Applications, Vol. 4, 2012, pp. 29-40. doi:10.4236/jilsa.2012.41003

[41] G. Serpen and M. Riesen “Knowledge Discovery for Query Formulation for Validation of a Bayesian Belief Network,” Journal of Intelligent Learning Systems and Applications, Vol. 2, No. 3, 2010, pp. 156-166

[1] Z. Pawlak, “Rough Sets,” International Journal of Parallel Programming, Vol. 11, No. 5, 1982, pp. 341-356. doi:10.1007/BF01001956

[2] X. Wang, E. C. C. Tsang, S. Zhao, D. Chen and S. Yeung, “Learning Fuzzy Rules from Fuzzy Samples Based on Rough Set Technique,” Information Sciences, Vol. 177, No. 15, 2007, pp. 4493-4514. doi:10.1016/j.ins.2007.04.010

[3] S. Zhao and E. C. C. Tsang, “On Fuzzy Approximation Operators in Attribute Reduction with Fuzzy Rough Sets,” Information Sciences, Vol. 178, No. 16, 2008, pp. 3163-3176.

[4] L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X

[5] Z. Bonikowski, “Algebraic Structures of Rough Sets,” In: W. Ziarko, Ed., Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer, London, 1994, pp. 243-247. doi:10.1007/978-1-4471-3238-7_29

[6] E. Bryniaski, “A Calculus of Rough Sets of The First Order,” Bulletin of the Polish Academy of Sciences, Vol. 16, 1989, pp. 71-77.

[7] Y. Y. Yao, “Constructive and Algebraic Methods of Theory of Rough Sets,” Information Sciences, Vol. 109, No. 1-4, 1998, pp. 21-47. doi:10.1016/S0020-0255(98)00012-7

[8] A. Skowron and J. Stepaniuk, “Tolerance Approximation Spaces,” Fund Information, Vol. 27, No. 2-3, 1996, pp. 245-253.

[9] E. F. Lashin, A. M. Kozae, A. A. Abo Khadra and T. Medhat, “Rough Set Theory for Topological Spaces,” International Journal of Approximate Reasoning, Vol. 40, No. 1-2, 2005, pp. 35-43. doi:10.1016/j.ijar.2004.11.007

[10] K. Qin and Z. Pei, “On the topological Properties of Fuzzy Rough Sets,” Fuzzy Sets and Systems, Vol. 151, No. 3, 2005, pp. 601-613. doi:10.1016/j.fss.2004.08.017

[11] A. Wasilewska, “Topological Rough Algebras,” In: T. Y. Lin and N. Cercone, Eds., Rough Sets and Data Mining, Kluwer Academic Publishers, Boston, 1997, pp. 425-441. doi:10.1007/978-1-4613-1461-5_21

[12] W. Zhu, “Topological Approaches to Covering Rough Sets,” Information Sciences, Vol. 177, No. 15, 2007, pp. 1499-1508. doi:10.1016/j.ins.2006.06.009

[13] T. J. Li, Y. Leung and W. X. Zhang, “Generalized Fuzzy Rough Approximation Operators Based on Fuzzy Coverings,” International Journal of Approximate Reasoning, Vol. 48, No. 3, 2008, pp. 836-856. doi:10.1016/j.ijar.2008.01.006

[14] R. Biswas, “On Rough Sets and Fuzzy Rough Sets,” Bulletin of the Polish Academy of Sciences, Vol. 42, 1992, pp. 343-349.

[15] G. Liu, “Generalized Rough Sets over Fuzzy Lattices,” Information Sciences, Vol. 178, No. 6, 2008, pp. 16511662. doi:10.1016/j.ins.2007.11.010

[16] A. M. Rolka and L. Rolka, “Fuzzy Rough Approximations of Process Data,” International Journal of Approximate Reasoning, Vol. 49, No. 2, 2008, pp. 301-315. doi:10.1016/j.ijar.2007.03.016

[17] M. Banerjee and S. K. Pal, “Roughness of a Fuzzy Set,” Information Sciences, Vol. 93, No. 3-4, 1995, pp. 235246. doi:10.1016/0020-0255(96)00081-3

[18] R. Biswas, “On Rough Fuzzy Sets,” Bulletin of the Polish Academy of Sciences, Vol. 42, 1994, pp. 352-355.

[19] D. Dubois and H. Prade, “Rough Fuzzy Sets and Fuzzy Rough Sets,” International Journal of General Systems, Vol. 17, No. 2-3, 1990, pp. 191-208. doi:10.1080/03081079008935107

[20] C. Degang, Y. Wenxia and Li Fachao, “Measures of General Fuzzy Rough Sets on a Probabilistic Space,” Information Sciences, Vol. 178, No. 16, 2008, pp. 3177-3187. doi:10.1016/j.ins.2008.03.020

[21] Z. Gong, B. Sun and D. Chen, “Rough Set Theory for the Interval-Valued Fuzzy Information Systems,” Information Sciences, Vol. 178, No. 8, 2008, pp. 1968-1985. doi:10.1016/j.ins.2007.12.005

[22] Y. Yang and C. Hinde, “A New Extension of Fuzzy Sets Using Rough Sets: R-Fuzzy Sets,” Information Sciences, Vol. 180, No. 3, 2010, pp. 354-365. doi:10.1016/j.ins.2009.10.004

[23] P. Bhattacharya and B. K. Lahiri, “Semi Generalized Closed Sets in Topology,” Indian Journal of Mathematics, Vol. 29, 1987, pp. 373-382.

[24] A. Skowron, “On Topology Information Systems,” Bulletin of the Polish Academy of Sciences, Vol. 3, 1989, pp. 87-90.

[25] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publishers, Boston, 1991.

[26] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, “On Pre-Continuous and Weak Pre Continuous Mappings,” Mathematical Physical and Engineering Sciences, Vol. 53, 1982, pp. 47-53.

[27] O. Najsted, “On Some Classes of Nearly Open Sets,” Pacific Journal of Mathematics, Vol. 15, 1965, pp. 961970.

[28] J. R. Munkres, “Topology, a First Course,” Prentice-Hall, Upper Saddle River, 1975.

[29] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, “?-Open Sets and ?-Continuous Mappings,” Bulletin of the Faculty of Science, Assiut University, Asyut, 1983.

[30] D. Andrijevic, “On b-Open Sets,” Matematicki Vesnik, Vol. 48, 1996, pp. 59-64.

[31] A. S. Mashhour, I. A. Hasanein and S. N. El Deeb, “A Note on α-Continuous and α-Open Mappings,” Acta Mathematica Hungarica, Vol. 41, No. 3-4, 1983, pp. 213218.

[32] J. Kortelainen, “On the Relationship between Modified Sets, Topological Spaces and Rough Sets,” Fuzzy Sets and Systems, Vol. 61, No. 1, 1994, pp. 91-95. doi:10.1016/0165-0114(94)90288-7

[33] Y. Y. Yao, “Two Views of the Theory of Rough Sets in Finite Universes,” International Journal of Approximate Reasoning, Vol. 15, No. 4, 1996, pp. 291-317. doi:10.1016/S0888-613X(96)00071-0

[34] Y. Y. Yao and T. Y. Lin, “Generalization of Rough Sets Using Modal Logics,” Intelligent Automation & Soft Computing, Vol. 2, 1996, pp. 103-120.

[35] Z. Pawlak, “On Rough Relations,” Bulletin of the Polish Academy of Sciences, Vol. 34, 1986, pp. 9-10.

[36] Z. Pawlak, “Rough Sets, Rough Relations and Rough Functions,” Bull of the Polish Academy of Sciences, Vol. 13, 1996, pp. 15-19.

[37] A. S. Salama, “Bitopological Rough Approximations with Medical Applications,” Journal of King Saud University (Science), Vol. 22, No. 3, 2010, pp. 177-183. doi:10.1016/j.jksus.2010.04.010

[38] J. Kelley, “General Topology,” Van Nostrand Company, New York, 1955.

[39] N. Levine, “Semi Open Sets and Semi Continuous Mappings in Topological Spaces,” American Mathematical Monthly, Vol. 70, No. 1, 1963, pp. 36-41. doi:10.2307/2312781

[40] M. H. F. Zarandi, et al., “A Fuzzy Expert System Architecture for Intelligent Tutoring Systems: A Cognitive Mapping Approach,” Journal of Intelligent Learning Systems and Applications, Vol. 4, 2012, pp. 29-40. doi:10.4236/jilsa.2012.41003

[41] G. Serpen and M. Riesen “Knowledge Discovery for Query Formulation for Validation of a Bayesian Belief Network,” Journal of Intelligent Learning Systems and Applications, Vol. 2, No. 3, 2010, pp. 156-166