Abstract
(0,1)-Qualitative approximations of fuzzy sets are studied by using the core and support of a fuzzy set. This setting naturally leads to three disjoint regions and an analysis based on a three-valued logic. This study combines both an algebra view and a logic view. From the algebra view, the mathematical definition of a (0,1)-approximation of fuzzy sets are given, and algebraic operations based on various t-norms and fuzzy implications are established. From the logic view, a non-classical three-valued logic is introduced. Corresponding to this new non-classical three-valued logic, the related origins of t-norms and fuzzy implications are examined.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chakrabarty, K., Biswas, R., Nanda, S.: Nearest ordinary set of a fuzzy set: a rough theoretic construction. Bulletin of the Polish Academy of Sciences, Technical Sciences 46, 105–114 (1998)
Chakrabarty, K., Biswas, R., Nanda, S.: Fuzziness in rough sets. Fuzzy Sets and Systems 110, 247–251 (2000)
Chanas, S.: On the interval approximation of a fuzzy number. Fuzzy Sets and Systems 122, 353–356 (2001)
Dubois, D., Prade, P.: A class of fuzzy measures based on triangular norms. International Journal of General Systems 8, 43–61 (1982)
Esteva, F., Godo, L.: Monoidal t-norm-based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)
Fodor, J.: Nilpotent minimum and related connectives for fuzzy logic. In: Proceedings of FUZZ-IEEE 1995, pp. 2077–2082 (1995)
Fodor, J., Keresztfalvi, T.: Nonstandard conjunctions and implications in fuzzy logic. International Journal of Approximate Reasoning 12, 69–84 (1995)
Giles, R.: The concept of grade of membership. Fuzzy Sets and Systems 25, 297–323 (1988)
Grzegorzewski, P.: Nearest interval approximation of a fuzzy number. Fuzzy Sets and Systems 130, 321–330 (2002)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets and Systems 143, 5–26 (2004)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey (1995)
Liu, G., Zhu, W.: The algebraic structures of generalized rough set theory. Information Sciences 178, 4105–4113 (2008)
Mi, J.S., Leung, Y., Zhao, H.Y., Feng, T.: Generalized fuzzy rough sets determined by a triangular norm. Information Sciences 178, 3203–3213 (2008)
Nguyen, H.T., Pedrycz, W., Kreinovich, V.: On approximation of fuzzy sets by crisp sets: from continuous control-oriented defuzzification to discrete decision making. In: Proceedings of International Conference on Intelligent Technologies, pp. 254–260 (2000)
Pawlak, Z., Skowron, A.: Rough membership functions, in: Yager, R.R., Fedrizzi, M. and Kacprzyk, J (Eds.). In: Advances in the Dempster-Shafer Theory of Evidence, pp. 251–271. John Wiley and Sons, New York (1994)
Pedrycz, W.: Shadowed sets: representing and processing fuzzy sets. IEEE Transactions on System, Man, and Cybernetics, Part B 28, 103–109 (1998)
Pedrycz, W.: Shadowed sets: bridging fuzzy and rough sets. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization a New Trend in Decision-making, pp. 179–199. Springer, Singapore (1999)
Pedrycz, W., Vukovich, G.: Investigating a relevance of fuzzy mappings. IEEE Transactions on System, Man, and Cybernetics, Part B 30, 249–262 (2000)
Wasilewska, A.: An Introduction to Classical and Non-Classical Logics. Suny, Stony Brook (2007)
Yao, Y.Y.: Interval-set algebra for qualitative knowledge representation. In: Proceedings of the Fifth International Conference on Computing and Information, pp. 370–374 (1993)
Yao, Y.Y.: Combination of rough and fuzzy sets based on α-level sets. In: Lin, T.Y., Cercone, N. (eds.) Rough Sets and Data Mining: Analysis for Imprecise Data, pp. 301–321. Kluwer Academic Publishers, Boston (1997)
Yao, Y.Y.: A comparative study of fuzzy sets and rough sets. Information Sciences 109, 227–242 (1998)
Yao, Y.Y.: Decision-theoretic rough set models. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślȩzak, D. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 1–12. Springer, Heidelberg (2007)
Yao, Y.Y.: Probabilistic rough set approximations. International Journal of Approximation Reasoning 49, 255–271 (2008)
Yao, Y.Y., Zhao, Y.: Attribute reduction in decision-theoretic rough set models. Information Sciences 178, 3356–3373 (2008)
Yao, Y.Y.: Crisp set approximations of fuzzy sets: a decision-theoretic rough sets perspective. Fuzzy Sets and Systems (2008)
Zhang, X.H.: Fuzzy Logics and Algebraic Analysis. Science Press, Beijing (2008)
Zhang, X.H., Fan, X.S.: Pseudo-BL algebras and pseudo-effect algebras. Fuzzy Sets and Systems 159(1), 95–106 (2008)
Zhu, W.: Relationship between generalized rough sets based on binary relation and covering. Information Sciences (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhang, X., Yao, Y., Zhao, Y. (2010). Qualitative Approximations of Fuzzy Sets and Non-classical Three-Valued Logics (I). In: Yu, J., Greco, S., Lingras, P., Wang, G., Skowron, A. (eds) Rough Set and Knowledge Technology. RSKT 2010. Lecture Notes in Computer Science(), vol 6401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16248-0_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-16248-0_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16247-3
Online ISBN: 978-3-642-16248-0
eBook Packages: Computer ScienceComputer Science (R0)