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Complex neutrosophic set

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Abstract

Complex fuzzy sets and complex intuitionistic fuzzy sets cannot handle imprecise, indeterminate, inconsistent, and incomplete information of periodic nature. To overcome this difficulty, we introduce complex neutrosophic set. A complex neutrosophic set is a neutrosophic set whose complex-valued truth membership function, complex-valued indeterminacy membership function, and complex-valued falsehood membership functions are the combination of real-valued truth amplitude term in association with phase term, real-valued indeterminate amplitude term with phase term, and real-valued false amplitude term with phase term, respectively. Complex neutrosophic set is an extension of the neutrosophic set. Further set theoretic operations such as complement, union, intersection, complex neutrosophic product, Cartesian product, distance measure, and δ-equalities of complex neutrosophic sets are studied here. A possible application of complex neutrosophic set is presented in this paper. Drawbacks and failure of the current methods are shown, and we also give a comparison of complex neutrosophic set to all such methods in this paper. We also showed in this paper the dominancy of complex neutrosophic set to all current methods through the graph.

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Acknowledgments

We are very thankful to Prof. Dr. Jie Lu of University of Technology Sydney, UTS Australia, for her valuable comments and suggestions. We are also very thankful to the reviewers for their comments and suggestions which improved this work.

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Authors and Affiliations

  1. Department of Mathematics, Quaid-e-Azam University, Islamabad, 45320, Pakistan

    Mumtaz Ali

  2. University of New Mexico, 705 Gurley Ave., Gallup, NM, 87301, USA

    Florentin Smarandache

Authors
  1. Mumtaz Ali
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  2. Florentin Smarandache
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Correspondence to Mumtaz Ali.

Appendix

Appendix

Comparison of complex neutrosophic sets to fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, complex fuzzy sets, and complex intuitionistic fuzzy sets is listed below (Table 1).

Table 1 Comparison of complex neutrosophic sets to the current approaches
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Ali, M., Smarandache, F. Complex neutrosophic set. Neural Comput & Applic 28, 1817–1834 (2017). https://doi.org/10.1007/s00521-015-2154-y

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