Advertisement

Type-1 Fuzzy Sets and Fuzzy Logic

  • Jerry M. Mendel
Chapter

Abstract

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap.  3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

References

  1. Aisbett, J., J.T. Rickard, and D.G. Morgenthaler. 2010. Type-2 fuzzy sets as functions on spaces. IEEE Trans on Fuzzy Systems 18: 841–844.CrossRefGoogle Scholar
  2. Allendoerfer, C.B., and C.O. Oakley. 1955. Principles of mathematics. New York: McGraw-Hill.zbMATHGoogle Scholar
  3. Arabi, B.N., N. Kehtarnavaz, and C. Lucas. 2001. Restrictions imposed by the fuzzy extension of relations and functions. Journal of Intelligent and Fuzzy Systems 11: 9–22.Google Scholar
  4. Bezdek, J.C. 1981. Pattern recognition with fuzzy objective function algorithms. New York: Plenum.CrossRefzbMATHGoogle Scholar
  5. Bezdek, J., and S.K. Pal. 1992. Fuzzy models for pattern recognition. New York: IEEE Press.Google Scholar
  6. Blanchard, N. 1982. Cardinal and ordinal theories about fuzzy sets. In Fuzzy Information and Decision Process, ed. M. M. Gupta and E. Sanchez, pp. 149–157, Amsterdam.Google Scholar
  7. Bonissone, P. P. and K. S. Decker. 1986. Selecting uncertainty calculi and granularity: An experiment in trading off precision and complexity. In Uncertainty in artificial intelligence, ed. L. N. Kanal and J. F. Lemmer, pp. 217–247, Amsterdam.Google Scholar
  8. Cheeseman, P. 1988. An inquiry into computer understanding. Computational Intelligence 4: 57–142 (with 22 commentaries/replies).Google Scholar
  9. Cox, E.A. 1994. The fuzzy systems handbook. Cambridge, MA: AP Professional.zbMATHGoogle Scholar
  10. Cox, E. A. 1992. Fuzzy fundamentals. IEEE Spectrum, pp. 58–61, Oct. 1992.Google Scholar
  11. Dubois, D., and H. Prade. 1980. Fuzzy sets and systems: Theory and applications. NY: Academic Press.zbMATHGoogle Scholar
  12. Dubois, D., and H. Prade. 1985. Fuzzy cardinality and the modeling of imprecise quantification. Fuzzy Sets and Systems 16: 199–230.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dutta, P. H. Boruah and T. Ali. 2011. Fuzzy arithmetic with and without using the α-cut method: A comparative study. International Journal of Latest Trends in Computing 2(1): 99–107 (E-ISSN: 2045-5364).Google Scholar
  14. Edwards, W.F. 1972. Likelihood. London: Cambridge Univ. Press.zbMATHGoogle Scholar
  15. Fan, J.L., W.X. Xie, and J. Pei. 1999. Subsethood measure: new definitions. Fuzzy Sets and Systems 106: 201–209.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gottwald, S. 1980. A note on fuzzy cardinals. Kybernetika 16: 156–158.MathSciNetzbMATHGoogle Scholar
  17. He, Q., H.-X. Li, C.L.P. Chen, and E.S. Lee. 2000. Extension principles and fuzzy set categories. Computers and Mathematics with Applications 39: 45–53.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Horikawa, S., T. Furahashi and Y. Uchikawa. 1992. On fuzzy modeling using fuzzy neural networks with back-propagation algorithm. IEEE Transaction on Neural Networks 3: 801–806.Google Scholar
  19. Jaccard, P. 1908. Nouvelles recherches sur la distribution florale. Bulletin de la Societe de Vaud des Sciences Naturelles 44: 223, 1908.Google Scholar
  20. Jang, J.-S.R. 1992. Self-learning fuzzy controllers based on temporal back-propagation. IEEE Transaction on Neural Networks 3: 714–723.CrossRefGoogle Scholar
  21. Jang, L.-C., and D. Ralescu. 2001. Cardinality concepts for type-two fuzzy sets. Fuzzy Sets and Systems 118: 479–487.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Jang, J.-S. R., C-T. Sun and E. Mizutani. 1997. Neuro-fuzzy and soft-computing, Prentice-Hall, Upper Saddle River, NJ.Google Scholar
  23. Karnik, N.N., and J.M. Mendel. 2001. Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122: 327–348.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Karnik, N. N. and J. M. Mendel. 1998. An introduction to type-2 fuzzy logic systems, USC-SIPI Report #418, Univ. of Southern Calif., Los Angeles, CA, June 1998. This can be accessed at: http://sipi.usc.edu/research; then choose “sipi technical reports/418.
  25. Kaufmann, A. 1977. Introduction a la theorie des sous-ensembles flous, complement et nouvelles applications, vol. 4. Paris: Masson.zbMATHGoogle Scholar
  26. Klement, E. P. 1982. On the cardinality of fuzzy sets. Proceedings of 6th European meeting on cybernetics and systems research, Vienna, pp. 701–704.Google Scholar
  27. Klir, G.J., and T.A. Folger. 1988. Fuzzy sets, uncertainty, and information. Englewood Cliffs, NJ: Prentice Hall.zbMATHGoogle Scholar
  28. Klir, G.J., and B. Yuan. 1995. Fuzzy sets and fuzzy logic: Theory and applications. Upper Saddle River, NJ: Prentice Hall.zbMATHGoogle Scholar
  29. Kosko, B. 1990. Fuzziness versus probability. International Journal of General Systems 17: 211–240.CrossRefzbMATHGoogle Scholar
  30. Kosko, B. 1992. Neural network and fuzzy systems, a dynamical systems approach to machine intelligence. Englewood Cliffs, NJ: Prentice-Hall.zbMATHGoogle Scholar
  31. Kosko. 1986. Fuzzy entropy and conditioning. Information Sciences 40: 165–174.Google Scholar
  32. Kreinovich, V. 2008. Relations between interval computing and soft computing. In Processing with interval and soft computing, ed. A. de Korvin and V. Kreinovich, 75–97. Springer, London.Google Scholar
  33. Larsen, P. M. 1980. Industrial applications of fuzzy logic control. International Journal Man-Machine Studies 12: 3–10.Google Scholar
  34. Laviolette, M., and J.W. Seaman Jr. 1994. The efficacy of fuzzy representations and uncertainty. IEEE Transaction on Fuzzy Systems 2: 4–15.CrossRefGoogle Scholar
  35. Lin, C.-T., and C.S.G. Lee. 1996. Neural fuzzy systems: A neuro-fuzzy synergism to intelligent systems. Upper Saddle River, NJ: Prentice-Hall PTR.Google Scholar
  36. Lindley, B. Y. 1982. Scoring rules and the inevitability of probability. International Statistical Review 50: 1–26 (with 7 commentaries/replies).Google Scholar
  37. De Luca, A., and S. Termini. 1972. A definition of non-probabilistic entropy in the setting of fuzzy sets theory. Information and Computation 20: 301–312.MathSciNetzbMATHGoogle Scholar
  38. Macvicar-Whelen, P. J. 1978. Fuzzy sets, the concept of height, and the hedge ‘very’. IEEE Transaction on Systems, Man, and Cybernetics SMC-8: 507–511.Google Scholar
  39. Mamdani, E.H. 1974. Applications of fuzzy algorithms for simple dynamic plant. Proceedings of the IEEE 121: 1585–1588.Google Scholar
  40. Mendel, J.M. 1995a. Fuzzy logic systems for engineering: A tutorial. IEEE Proceedings 83: 345–377.CrossRefGoogle Scholar
  41. Mendel, J.M. 1995b. Lessons in estimation theory for signal processing. Prentice-Hall PTR, Englewood Cliffs, NJ: Communications and Control.zbMATHGoogle Scholar
  42. Mendel, J.M. 2007. Computing with words: Zadeh, Turing, Popper and Occam. IEEE Computational Intelligence Magazine 2: 10–17.CrossRefGoogle Scholar
  43. Mendel, J.M. 2015. Type-2 fuzzy sets and systems: A retrospective. Informatik Spektrum 38 (6): 523–532.CrossRefGoogle Scholar
  44. Mendel, J.M., H. Hagras, W.-W. Tan, W.W. Melek, and H. Ying. 2014. Introduction to type-2 fuzzy logic control. Hoboken, NJ: John Wiley and IEEE Press.CrossRefzbMATHGoogle Scholar
  45. Mendel, J.M., and D. Wu. 2010. Perceptual computing: Aiding people in making subjective judgments. Hoboken, NJ: Wiley and IEEE Press.CrossRefGoogle Scholar
  46. Nguyen, H.T. 1978. A note on the extension principle for fuzzy sets. Journal of Mathematical Analysis and Applications 64: 369–380.MathSciNetCrossRefzbMATHGoogle Scholar
  47. Nguyen, H. T. and V. Kreinovich. 2008. Computing degrees of subsethood and similarity for interval-valued fuzzy sets: Fast algorithms. In Proceedings of 9th international conference on intelligent technologies in tech’08, pp. 47–55, Samui, Thailand, Oct. 2008.Google Scholar
  48. Rajati, M.R., and J.M. Mendel. 2013. Novel weighted averages versus normalized sums in computing with words. Information Sciences 235: 130–149.MathSciNetCrossRefzbMATHGoogle Scholar
  49. Rudin, W. 1966. Real and complex analysis. New York: Mc-Graw Hill.zbMATHGoogle Scholar
  50. Ruspini, E. 1969. A new approach to clustering. Information Control 15: 22–32.CrossRefzbMATHGoogle Scholar
  51. Schmucker, K.S. 1984. Fuzzy set, natural language computations, and risk analysis. Rockville, MD: Computer Science Press.zbMATHGoogle Scholar
  52. Wang, L.-X. 1997. A course in fuzzy systems and control. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  53. Wang, L.-X., and J.M. Mendel. 2016. Fuzzy opinion networks: A mathematical framework for the evolution of opinions and their uncertainties across social networks. IEEE Transaction on Fuzzy Systems 24: 880–905.CrossRefGoogle Scholar
  54. Wang, L.-X. and J. M. Mendel. 1992a. Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans. on Neural Networks 3: 807–813.Google Scholar
  55. Wang, L.-X. and J. M. Mendel. 1992b. Back-propagation of fuzzy systems as non-linear dynamic system identifiers. Proceedingas of IEEE intternational conference on fuzzy systems, 1409–1418, San Diego, CA.Google Scholar
  56. Wei, S-H. and S-M. Chen. 2009. Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Systems with Applications 36: 2285–2299.Google Scholar
  57. Wu, D., and J.M. Mendel. 2007. Uncertainty measures for interval type-2 fuzzy sets. Information Sciences 177: 5378–5393.MathSciNetCrossRefzbMATHGoogle Scholar
  58. Wygralak, M. 1983. A new approach to the fuzzy cardinality of finite fuzzy sets. Busefal 15: 72–75.zbMATHGoogle Scholar
  59. Wygralak, M. 2003. Cardinalities of fuzzy sets. Heidelberg: Springer.CrossRefzbMATHGoogle Scholar
  60. Yager, R.R. 1986. A characterization of the fuzzy extension principle. Journal Fuzzy Sets and Systems 18: 205–217.MathSciNetCrossRefzbMATHGoogle Scholar
  61. Yager, R.R., and D.P. Filev. 1994. Essentials of fuzzy modeling and control. New York: Wiley.Google Scholar
  62. Young, V.R. 1996. Fuzzy subsethood. Fuzzy Sets and Systems 77: 371–384.MathSciNetCrossRefzbMATHGoogle Scholar
  63. Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338–353.MathSciNetCrossRefzbMATHGoogle Scholar
  64. Zadeh, L.A. 1971. Similarity relations and fuzzy orderings. Information Sciences 3 (2): 177–200.MathSciNetCrossRefzbMATHGoogle Scholar
  65. Zadeh, L.A. 1972. A fuzzy-set-theoretic interpretation of iinguistic hedges. Journal of Cybernetics 2: 4–34.MathSciNetCrossRefGoogle Scholar
  66. Zadeh, L.A. 1981. Possibility theory and soft data analysis. In Mathematical frontiers of the social and policy sciences, ed. L. Cobb, and R.M. Thrall, 69–129. CO: Westview Press, Boulder.Google Scholar
  67. Zadeh, L. A. 1973. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transaction on Systems, Man, and Cybernetics SMC-3: 28–44.Google Scholar
  68. Zadeh, L. A. 1975. The concept of a linguistic variable and Its application to approximate reasoning–1. Information Sciences 8: 199–249.Google Scholar
  69. Zadeh, L. A. 1999. From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions. IEEE Transaction on Circuits and Systems–I: Fundamental Theory and Applications, 4: 105–119.Google Scholar
  70. Zimmermann, H.J. 1991. Fuzzy set theory and its applications, 2nd ed. Boston, MA: Kluwer Academic Publ.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

Personalised recommendations