Abstract

Fuzzy logic system studies reasoning systems in which the design of precision and deception are considered in a graded fashion, in contrast with classical mathematics where only absolutely true statements are considered. Whereas, Axiomatic fuzzy logic system facilitates a significant step on how to transform the information within databases into the membership functions and their fuzzy logic operations, by taking both the fuzziness and randomness into account. In this paper, various notations and illustrations of fuzzy concepts and coherence membership functions have been studied and analyzed under the framework of Axiomatic Fuzzy set theory. Various examples are illustrated for every concept by considering the hypothetical data.

References

• X.D. Liu, The Fuzzy Theory Based on AFS Algebras and AFSStructure, Journal of Mathematical Analysis and Applications217 (1998) 459–478.
• X.D. Liu, W. Pedrycz, Q. Zhang, Axiomatic Fuzzy Sets Logic,IEEE International Conference on Fuzzy Systems 1 (2003) 55–60.
• X.D. Liu, The Topology on AFS Algebra and AFS Structure,Journal of Mathematical Analysis and Applications 217 (1998)479 – 489.
• X.D. Liu, The Fuzzy Sets and Systems Based on AFS Structure,EI algebra and EII algebra, Fuzzy Sets and Systems 95 (1998)179 –188.
• X.D. Liu, W. Pedrycz, The Development of Fuzzy DecisionTrees in the Framework of Axiomatic Fuzzy Set Logic, Applied Soft Computing 7 (2007) 325–342.
• X.D. Liu, W. Wang, T. Chai, The Fuzzy Clustering Analysis Based on AFS Theory, IEEE Transactions on Systems, Man and Cybernetics: Part B 35 (5) (2005)
• Yan Ren, Mingli Song, and Xiaodong Liu, New Approaches to the Fuzzy Clustering Via AFS Theory, International Journal of Information and Systems Sciences, Volume 3, Number 2, Pages 307-325(2007)
• X.D. Liu, W. Liu, The Framework of Axiomatic Fuzzy Sets Based Fuzzy Classifiers, Journal of Industrial Management Optimization 4 (3) (2008) 581–609.
• L. Wang, X. Liu, Concept Analysis via Rough Set and AFS Algebra, Information Sciences 178 (2008) 4125–4137.
• X.D. Liu, W. Pedrycz, T.Y. Chai, M.L. Song, The Development of Fuzzy Rough Sets with The Use of Structures and Algebras of Axiomatic Fuzzy Sets, IEEE Transactions on Knowledge and Data Engineering 21 (3) (2009) 443–462.
• X.D. Liu, Q.L. Zhang, The Fuzzy Cognitive Maps Based on AFS Fuzzy Logic, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 11 (2004) 787–796.
• X.D. Liu, W.Q. Liu, Credit Rating Analysis with AFS Fuzzy Logic, Lecture Notes in Computer Science (LNCS) 3612 (2005) 1198–1204.
• X.D. Liu, W. Pedrycz, Axiomatic Fuzzy Set Theory and Its Applications, Springer Verlag, Heidelberg, Germany, 2009.
• X.D. Liu, T.Y. Chai, W. Wang, W.Q. Liu, Approaches to the representations and logic operations for fuzzy concepts in the framework of axiomatic fuzzy set theory I, II, Information Sciences 177 (2007) 1007–1026, 1027–1045.
• X. Liu, The Structure of Fuzzy Matrices, Journal of Fuzzy Mathematics 2 (1994) 311–325.
• X. Liu, A new mathematical axiomatic system of fuzzy sets and systems, Journal of Fuzzy Mathematics 3 (1995) 559–560.
• X.D. Liu, Two algebra structures of AFS structure, Journal of Fuzzy Mathematics 3 (1995) 561–562.
• X. Liu, Q. Zhang, AFS fuzzy logic and its applications toFuzzy information processing, Dongbei Daxue Xuebao 23 (4) (2002) 321– 323 (in Chinese).
• X. Liu, Q. Zhang, The EI algebra representations of fuzzyconcept, Fuzzy Systems and Mathematics 16 (2) (2002) 27–35.
• X. Liu, K. Zhu, H. Huang, The representations of fuzzy concepts based on the fuzzy matrix theory and the AFS theory, in: Proceedings of IEEE International Symposium on Intelligent Control, Houston, TX, USA, 2003, pp. 1006–1011.
• K.H. Kim, Boolean Matrix Theory and Applications, Marcel Dekker, New York, 1982.
• X. Liu, X. Feng, Witold Pedrycz, Extraction of fuzzy rules from fuzzy decision trees: on Axiomatic fuzzy sets approach, Data& knowledge Engineering 84 (2013), 1-25.
• Y. Zhang, D. Liang, S. Tong, On AFS algebra Part I, Information Sciences 167 (2004) 263–286.
• Y. Zhang, D. Liang, S. Tong, On AFS algebra Part II, Information Sciences 167 (2004) 287–303.