A Study of Operators on Fuzzy Sets

Alhaji Jibril Alkali; Sylvanus Kupongoh Samaila

doi:doi:10.11648/j.ml.20210702.13

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A Study of Operators on Fuzzy Sets

Alhaji Jibril Alkali, Sylvanus Kupongoh Samaila

Published in Mathematics Letters (Volume 7, Issue 2)

Received: 4 April 2021 Accepted: 20 May 2021 Published: 15 June 2021

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Abstract

Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using t-norm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.

Published in	Mathematics Letters (Volume 7, Issue 2)
DOI	10.11648/j.ml.20210702.13
Page(s)	30-36
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Fuzzy Sets, Complement Operators, Union Operators, Intersection Operators, T-norm, T-conorm, T1FSs, T2FSs

References

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[3]	Castillo, O. and Aguilar, T. (2019). Type-2 Fuzzy logic in control of nonsmooth systems: Theoretical concepts and Applications. Springer Nature, Switzerland AG, Switzerland.
[4]	Chang S. and Zadeh, L. A. (1972). On Fuzzy Mappings and Control. IEEE Transactions on systems Man and cybernetics, SMC-2 (1) 30-34.
[5]	Chen, G. and Pham, T. (2001). Introduction to fuzzy sets, fuzzy logic and fuzzy control systems CRC Press, New York.
[6]	Dubois, D. and Prade, H. (1980). Fuzzy stes and systems: Theory and Applications. Academic Press, INC, New York.
[7]	Dubois, D. and Prade, H. (2000). Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, New York.
[8]	H. Garg,, H., Gwak, J., Mahmood, T. and Ali, Z. (2020). Power Aggregation Operators and VIKOR methods for a complex q-Rung Orthopair Fuzzy Sets and Thier Applications mathematics, 8, 538.
[9]	Gen, M., Tsujimura, Y., and Zheng, D. (1997). An application of fuzzy set theory to inventory control model. Computers Ind. Engng 33, 553-556.
[10]	Hamrawi, H. (2011). Type-2 fuzzy alpha-cuts PhD Thesis, De Mentfort University, London, (2011).
[11]	Karnik, N., and Mendel, J. (2001a). Centroid of a type-2 fuzzy set. Information Sciences 132, 195-220.
[12]	Karnik, N. and Mendel, J. (2001b). OPerations on Type-2 Fuzzy sets. Fuzzy Sets and Systems 122, 327-348.
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[19]	H. J. Zimmermann, H-J (1996). Fuzzy set theory and its Applications 3rd Edition. Kluwer Academic Publishing Group, London.
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[21]	Zimmermann, H-J, (2001). Fuzzy set theory and its Applications 4th Edition. Springer Science + Business Media LLC, New York.

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APA Style

Alhaji Jibril Alkali, Sylvanus Kupongoh Samaila. (2021). A Study of Operators on Fuzzy Sets. Mathematics Letters, 7(2), 30-36. https://doi.org/10.11648/j.ml.20210702.13

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ACS Style

Alhaji Jibril Alkali; Sylvanus Kupongoh Samaila. A Study of Operators on Fuzzy Sets. Math. Lett. 2021, 7(2), 30-36. doi: 10.11648/j.ml.20210702.13

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AMA Style

Alhaji Jibril Alkali, Sylvanus Kupongoh Samaila. A Study of Operators on Fuzzy Sets. Math Lett. 2021;7(2):30-36. doi: 10.11648/j.ml.20210702.13

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@article{10.11648/j.ml.20210702.13,
  author = {Alhaji Jibril Alkali and Sylvanus Kupongoh Samaila},
  title = {A Study of Operators on Fuzzy Sets},
  journal = {Mathematics Letters},
  volume = {7},
  number = {2},
  pages = {30-36},
  doi = {10.11648/j.ml.20210702.13},
  url = {https://doi.org/10.11648/j.ml.20210702.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210702.13},
  abstract = {Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using t-norm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.},
 year = {2021}
}

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TY - JOUR
T1 - A Study of Operators on Fuzzy Sets
AU - Alhaji Jibril Alkali
AU - Sylvanus Kupongoh Samaila
Y1 - 2021/06/15
PY - 2021
N1 - https://doi.org/10.11648/j.ml.20210702.13
DO - 10.11648/j.ml.20210702.13
T2 - Mathematics Letters
JF - Mathematics Letters
JO - Mathematics Letters
SP - 30
EP - 36
PB - Science Publishing Group
SN - 2575-5056
UR - https://doi.org/10.11648/j.ml.20210702.13
AB - Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using t-norm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.
VL - 7
IS - 2
ER -

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Author Information

Alhaji Jibril Alkali

Department of Mathematics, Ahmadu Bello University Zaria, Zaria, Nigeria
Sylvanus Kupongoh Samaila

Department of Mathematics and Statistics, Federal University Otuoke, Otuoke, Nigeria

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APA Style

Alhaji Jibril Alkali, Sylvanus Kupongoh Samaila. (2021). A Study of Operators on Fuzzy Sets. Mathematics Letters, 7(2), 30-36. https://doi.org/10.11648/j.ml.20210702.13

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ACS Style

Alhaji Jibril Alkali; Sylvanus Kupongoh Samaila. A Study of Operators on Fuzzy Sets. Math. Lett. 2021, 7(2), 30-36. doi: 10.11648/j.ml.20210702.13

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AMA Style

Alhaji Jibril Alkali, Sylvanus Kupongoh Samaila. A Study of Operators on Fuzzy Sets. Math Lett. 2021;7(2):30-36. doi: 10.11648/j.ml.20210702.13

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@article{10.11648/j.ml.20210702.13,
  author = {Alhaji Jibril Alkali and Sylvanus Kupongoh Samaila},
  title = {A Study of Operators on Fuzzy Sets},
  journal = {Mathematics Letters},
  volume = {7},
  number = {2},
  pages = {30-36},
  doi = {10.11648/j.ml.20210702.13},
  url = {https://doi.org/10.11648/j.ml.20210702.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210702.13},
  abstract = {Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using t-norm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.},
 year = {2021}
}

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