A new definition of the fuzzy set

Andrzej Piegat

International Journal of Applied Mathematics and Computer Science (2005)

  • Volume: 15, Issue: 1, page 125-140
  • ISSN: 1641-876X

Abstract

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The present fuzzy arithmetic based on Zadeh's possibilistic extension principle and on the classic definition of a fuzzy set has many essential drawbacks. Therefore its application to the solution of practical tasks is limited. In the paper a new definition of the fuzzy set is presented. The definition allows for a considerable fuzziness decrease in the number of arithmetic operations in comparison with the results produced by the present fuzzy arithmetic.

How to cite

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Piegat, Andrzej. "A new definition of the fuzzy set." International Journal of Applied Mathematics and Computer Science 15.1 (2005): 125-140. <http://eudml.org/doc/207721>.

@article{Piegat2005,
abstract = {The present fuzzy arithmetic based on Zadeh's possibilistic extension principle and on the classic definition of a fuzzy set has many essential drawbacks. Therefore its application to the solution of practical tasks is limited. In the paper a new definition of the fuzzy set is presented. The definition allows for a considerable fuzziness decrease in the number of arithmetic operations in comparison with the results produced by the present fuzzy arithmetic.},
author = {Piegat, Andrzej},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {fuzzy set theory; possibility; fuzzy arithmetic; fuzzy sets; membership function determination},
language = {eng},
number = {1},
pages = {125-140},
title = {A new definition of the fuzzy set},
url = {http://eudml.org/doc/207721},
volume = {15},
year = {2005},
}

TY - JOUR
AU - Piegat, Andrzej
TI - A new definition of the fuzzy set
JO - International Journal of Applied Mathematics and Computer Science
PY - 2005
VL - 15
IS - 1
SP - 125
EP - 140
AB - The present fuzzy arithmetic based on Zadeh's possibilistic extension principle and on the classic definition of a fuzzy set has many essential drawbacks. Therefore its application to the solution of practical tasks is limited. In the paper a new definition of the fuzzy set is presented. The definition allows for a considerable fuzziness decrease in the number of arithmetic operations in comparison with the results produced by the present fuzzy arithmetic.
LA - eng
KW - fuzzy set theory; possibility; fuzzy arithmetic; fuzzy sets; membership function determination
UR - http://eudml.org/doc/207721
ER -

References

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  1. Bezdek J. (1993): ditorial, fuzzy models - What are they, and why?.- IEEE Trans. Fuzzy Syst., Vol. 1, No. 1, pp. 1-6. 
  2. Driankov D., Hellendorn H. and Reinfrank M. (1993): An Introduction to Fuzzy Control. - Berlin:Springer. 
  3. Dubois D. and Prade H. (1988): Possibility Theory. - New York: Plenum Press. Zbl0645.68108
  4. Dubois D. and Prade H. (1996): An introduction to fuzzy systems. - Int. J. Appl. Math. Comput. Sci., Vol. 6, No. 3, pp. 485-503. 
  5. Dubois D. and Prade H. (1997): The three semantics of fuzzy sets. - Fuzzy Sets Syst., Vol. 90, No. 2, pp.141-150. Zbl0919.04006
  6. Kaufmann A. and Gupta M.M. (1991): Introduction to Fuzzy Arithmetic.- New York: Van Nostrand Reinhold. Zbl0754.26012
  7. Klir G.J. (1997): Fuzzy arithmetic with requisite constraints. - Fuzzy Sets Syst., Vol. 91, pp. 165-175. Zbl0920.04007
  8. Klir G.J. and Folger T.A. (1988): Fuzzy Sets, Uncertainty, and Information.- Englewood Cliffs: Prentice Hall. Zbl0675.94025
  9. Kosiński W., Prokopowicz P. and Ślęzak D. (2003): Ordered fuzzy numbers.- Bull. Polish Acad. Sci. Math., Vol. 51, No. 3, pp. 329-341. Zbl1102.03310
  10. Pearsal J. (Ed.) (1999): The New Oxford Dictionary of English.- Oxford: Oxford University Press. 
  11. Piegat A. (2001): Fuzzy Modeling and Control. - Heidelberg, New York: Springer-Verlag. Zbl0976.93001
  12. Piegat A. (2004): Is fuzzy evaluation a measurement? In: Soft Computing, Tools, Techniques and Applications (P. Grzegorzewski, M. Krawczakand S. Zadrożny, Eds.). - Warszawa:Akademicka Oficyna Wydawnicza EXIT, pp. 257-266. 
  13. Piegat A. (2005a): On practical problems with explanation of the difference between possibility and probability. - Contr. Cybern., (accepted for publication in No. 2 in 2005). 
  14. Piegat A. (2005b): Informative value of the possibilistic extension principle, In: Enhanced Methods in Computer Security, Biometric and Artificial Intelligence Systems (J. Pejas and A. Piegat, Eds.).- New York: Springer Science Business Media, Inc., pp. 301-310. 
  15. Yager R.R. and Filev D.P. (1994): Essentials of Fuzzy Modeling and Control. - London: Wiley. 
  16. Zadeh L.A. (1965): Fuzzy Sets. - Inf. Contr., Vol. 8, No. 3, pp. 338-353. Zbl0139.24606
  17. Zadeh L.A. (1978): Fuzzy sets as a basis for a theory of possibility. - Fuzzy Sets Syst., Vol. 1, No. 28, pp. 3-28. Zbl0377.04002
  18. Zadeh L.A. (2002): From computing with numbers to computing with words - From manipulation of measurements to manipulation of perceptions. - Int. J. Appl. Math. Comput. Sci., Vol. 12, No. 3, pp. 307-324. Zbl1062.68583
  19. Zimmermann H.J. (1996): Fuzzy Set Theory. - Boston: Kluwer. Zbl0845.04006

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