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Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties

Year 2019, Volume: 7 Issue: 2, 218 - 224, 15.10.2019

Nurettin Irmak Naime Demirtaş

https://doi.org/10.36753/mathenot.634513
Cited By: 1

Abstract

In this paper, we combine the important concepts which are Fuzzy numbers and Fibonacci, Lucas numbers. We introduce the concepts of Fuzzy Fibonacci and Fuzzy Lucas numbers by this combination. By this motivation, we provide a bridge between the areas Fuzzy sets and number theory. Afterwards, we generalize their well-known properties by the definitions of Fuzzy Fibonacci and Lucas numbers.

Keywords

Fuzzy numbers Fuzzy Fibonacci numbers Fuzzy Lucas numbers

References

  • \bibitem{Bandemer} Bandemer, H., Mathematics of Uncertainty: Ideas, Methods, Application Problems. Springer, New York, 2006. \bibitem{prade} Dubois, D. and Prade, H., Operations on Fuzzy Numbers,{\it Int. J. Systems Sci.,} 9-6, 613-626, 1978.
  • \bibitem{Dubois} Dubois D. and Prade H., Towards fuzzy differential calculus. {\it Fuzzy Sets Syst.} 8 (1982) 1–17(I), 105–116(II), 225–234(III).
  • \bibitem{Diamond} Diamond P. and Kloeden. P., Metric Spaces of Fuzzy Sets. World Scientific, Singapore, 1994.
  • \bibitem{Dubois2} Dubois D. and Prade H., Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.
  • \bibitem{Dubois3} Dubois D. and Prade H., Ranking fuzzy numbers in a setting of possibility theory. {\it Inf. Sci.} 30 (1983) 183–224.
  • \bibitem{Dubois4} Dubois D. and Prade H., Possibility Theory. An Approach to Computerized Processing of Uncertainty. Plenum Press, New York, 1988.
  • \bibitem{dubois5} Dubois D. and Prade H., (eds). Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series. Kluwer, Boston, 2000.
  • \bibitem{dubois6} Dubois D., Kerre E., Mesiar R. and Prade H., Fuzzy interval analysis. In: D. Dubois and H. Prade (eds), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series. Kluwer, Boston, 2000, pp. 483–581.
  • \bibitem{gao} Gao, S., Zhang, Z. and Cao, C., Multiplication Operation on Fuzzy Numbers,{\it Journal of Software,} 4-4, 331-338, 2009.
  • \bibitem{Goetschel} Goetschel R. and Voxman W., Elementary fuzzy calculus. {\it Fuzzy Sets Syst.} 18 (1986) 31–43.
  • \bibitem{Guerra} Guerra M.L. and Stefanini L., Approximate fuzzy arithmetic operations using monotonic interpolations. {\it Fuzzy Sets Syst}. 150 (2005) 5–33.
  • \bibitem{halici} Halıcı, S., On Fibonacci Quaterions, {\it Adv. Appl. Clifford Algebras}, 12 (2012), 321-327.
  • \bibitem{Kaufmann} Kaufmann, A. and Gupta, M.M., Introduction to Fuzzy Arithmetic – Theory and Applications. Van Nostrand Reinhold, New York, 1985.
  • \bibitem{klir} Klir, G.J., Uncertainty Analysis in Engineering and Science, Kluwer, Dordrecht, 1997.
  • \bibitem{Klir2} Klir, G.J. and Yuan B., Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ, 1995.
  • \bibitem{koshy} Koshy, T., Fibonacci and Lucas numbers with their identities, Wiley, New York, 2001.
  • \bibitem{Stefanini} Stefanini L., Sorini L. and Guerra M.L., Parametric representation of fuzzy numbers and application to fuzzy calculus. {\it Fuzzy Sets Syst.} 157 (2006) 2423–2455
  • \bibitem{Stefanini2} Stefanini L. and Sorini L., and Guerra M.L., Simulation of fuzzy dynamical systems using the LU-representation of fuzzy numbers. {\it Chaos Solitons Fractals} 29(3) (2006) 638–652.
  • \bibitem{zimmermann} Zimmermann H.-J., Fuzzy Set Theory and Its Applications, 4th edn. Kluwer, Dordrecht, 2001.
  • \bibitem{zadeh} Zadeh, L. A., Fuzzy Sets, {\it Information and Control,} 8-3, 338-353, 1965.

Year 2019, Volume: 7 Issue: 2, 218 - 224, 15.10.2019

Nurettin Irmak Naime Demirtaş

https://doi.org/10.36753/mathenot.634513
Cited By: 1

Abstract

References

  • \bibitem{Bandemer} Bandemer, H., Mathematics of Uncertainty: Ideas, Methods, Application Problems. Springer, New York, 2006. \bibitem{prade} Dubois, D. and Prade, H., Operations on Fuzzy Numbers,{\it Int. J. Systems Sci.,} 9-6, 613-626, 1978.
  • \bibitem{Dubois} Dubois D. and Prade H., Towards fuzzy differential calculus. {\it Fuzzy Sets Syst.} 8 (1982) 1–17(I), 105–116(II), 225–234(III).
  • \bibitem{Diamond} Diamond P. and Kloeden. P., Metric Spaces of Fuzzy Sets. World Scientific, Singapore, 1994.
  • \bibitem{Dubois2} Dubois D. and Prade H., Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.
  • \bibitem{Dubois3} Dubois D. and Prade H., Ranking fuzzy numbers in a setting of possibility theory. {\it Inf. Sci.} 30 (1983) 183–224.
  • \bibitem{Dubois4} Dubois D. and Prade H., Possibility Theory. An Approach to Computerized Processing of Uncertainty. Plenum Press, New York, 1988.
  • \bibitem{dubois5} Dubois D. and Prade H., (eds). Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series. Kluwer, Boston, 2000.
  • \bibitem{dubois6} Dubois D., Kerre E., Mesiar R. and Prade H., Fuzzy interval analysis. In: D. Dubois and H. Prade (eds), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series. Kluwer, Boston, 2000, pp. 483–581.
  • \bibitem{gao} Gao, S., Zhang, Z. and Cao, C., Multiplication Operation on Fuzzy Numbers,{\it Journal of Software,} 4-4, 331-338, 2009.
  • \bibitem{Goetschel} Goetschel R. and Voxman W., Elementary fuzzy calculus. {\it Fuzzy Sets Syst.} 18 (1986) 31–43.
  • \bibitem{Guerra} Guerra M.L. and Stefanini L., Approximate fuzzy arithmetic operations using monotonic interpolations. {\it Fuzzy Sets Syst}. 150 (2005) 5–33.
  • \bibitem{halici} Halıcı, S., On Fibonacci Quaterions, {\it Adv. Appl. Clifford Algebras}, 12 (2012), 321-327.
  • \bibitem{Kaufmann} Kaufmann, A. and Gupta, M.M., Introduction to Fuzzy Arithmetic – Theory and Applications. Van Nostrand Reinhold, New York, 1985.
  • \bibitem{klir} Klir, G.J., Uncertainty Analysis in Engineering and Science, Kluwer, Dordrecht, 1997.
  • \bibitem{Klir2} Klir, G.J. and Yuan B., Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ, 1995.
  • \bibitem{koshy} Koshy, T., Fibonacci and Lucas numbers with their identities, Wiley, New York, 2001.
  • \bibitem{Stefanini} Stefanini L., Sorini L. and Guerra M.L., Parametric representation of fuzzy numbers and application to fuzzy calculus. {\it Fuzzy Sets Syst.} 157 (2006) 2423–2455
  • \bibitem{Stefanini2} Stefanini L. and Sorini L., and Guerra M.L., Simulation of fuzzy dynamical systems using the LU-representation of fuzzy numbers. {\it Chaos Solitons Fractals} 29(3) (2006) 638–652.
  • \bibitem{zimmermann} Zimmermann H.-J., Fuzzy Set Theory and Its Applications, 4th edn. Kluwer, Dordrecht, 2001.
  • \bibitem{zadeh} Zadeh, L. A., Fuzzy Sets, {\it Information and Control,} 8-3, 338-353, 1965.
There are 20 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Nurettin Irmak

Naime Demirtaş

Publication Date October 15, 2019
Submission Date October 22, 2018
Acceptance Date October 7, 2019
Published in Issue Year 2019 Volume: 7 Issue: 2

Cite

APA Irmak, N., & Demirtaş, N. (2019). Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties. Mathematical Sciences and Applications E-Notes, 7(2), 218-224. https://doi.org/10.36753/mathenot.634513
AMA Irmak N, Demirtaş N. Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties. Math. Sci. Appl. E-Notes. October 2019;7(2):218-224. doi:10.36753/mathenot.634513
Chicago Irmak, Nurettin, and Naime Demirtaş. “Fuzzy Fibonacci and Fuzzy Lucas Numbers With Their Properties”. Mathematical Sciences and Applications E-Notes 7, no. 2 (October 2019): 218-24. https://doi.org/10.36753/mathenot.634513.
EndNote Irmak N, Demirtaş N (October 1, 2019) Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties. Mathematical Sciences and Applications E-Notes 7 2 218–224.
IEEE N. Irmak and N. Demirtaş, “Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties”, Math. Sci. Appl. E-Notes, vol. 7, no. 2, pp. 218–224, 2019, doi: 10.36753/mathenot.634513.
ISNAD Irmak, Nurettin - Demirtaş, Naime. “Fuzzy Fibonacci and Fuzzy Lucas Numbers With Their Properties”. Mathematical Sciences and Applications E-Notes 7/2 (October 2019), 218-224. https://doi.org/10.36753/mathenot.634513.
JAMA Irmak N, Demirtaş N. Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties. Math. Sci. Appl. E-Notes. 2019;7:218–224.
MLA Irmak, Nurettin and Naime Demirtaş. “Fuzzy Fibonacci and Fuzzy Lucas Numbers With Their Properties”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 2, 2019, pp. 218-24, doi:10.36753/mathenot.634513.
Vancouver Irmak N, Demirtaş N. Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties. Math. Sci. Appl. E-Notes. 2019;7(2):218-24.

Cited By

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