Some concepts on interval-valued refined neutrosophic sets and their applications
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ORIGINAL RESEARCH
Some concepts on interval‑valued refined neutrosophic sets and their applications Vakkas Uluçay1 Received: 6 March 2020 / Accepted: 29 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract An interval-valued refined neutrosophic set is simply an extension of interval neutrosophic set and refined neutrosophic set which can be used in statistics, game theory, engineering, and experimental science. In this study, we define the cut set and extension principle based on interval-valued refined neutrosophic sets which is a bridge between interval-valued refined neutrosophic sets and crisp sets. Also, we examine the properties of the cut sets and extension principle of interval-valued refined neutrosophic sets. Finally, according to the extension principle of the interval-valued refined neutrosophic sets, we introduce some algebraic operations over the interval-valued refined neutrosophic sets. Keywords Neutrosophic set · Interval-valued refined neutrosophic set · Cut set · Extension principle · Algebraic operation
1 Introduction In 1999, Smarandache (1998) proposed the concept of neutrosophic set (NS for short) by adding an independent indeterminacy-membership function which is a generalization of classic set, fuzzy set (Zadeh 1965), intuitionistic fuzzy set (Atanassov 1986) and so on. A NS is a set in which each element of the universe has respective degrees of truth, indeterminacy, and falsity, which lie in the nonstandard unit interval of ]− 0, 1+ [ This method represents an extension of the standard interval [0, 1] used for IFSs. The uncertainty presented here, (i.e., the indeterminacy factor) is independent of the truth and falsity values. This extended IFS theory to account for uncertain information.In NS, the indeterminacy is quantified explicitly and truth-membership (T), indeterminacy (I) membership, and false-membership (F) are completely independent and from scientific or engineering point of view, the NS operators need to be specified. Therefore, Wang et al. (2010) defined a single valued neutrosophic set (SVNS) and then provided the set theoretic operations and various properties of single valued neutrosophic sets and Wang et al. (2005) proposed the set theoretic operations on an instance of neutrosophic set is called interval valued * Vakkas Uluçay [email protected] 1
Department of Mathematics, Kilis 7 Aralık University, 79000 Kilis, Turkey
neutrosophic set (IVNS) which is more flexible and practical than NS. The works on single valued neutrosophic set (SVNS) and interval valued neutrosophic sets (IVNS) and their hybrid structure in theories and application have been progressing rapidly (e.g., Aslan et al. 2020; Broumi et al. 2015a; Broumi and Deli 2015; Broumi et al. 2014; Broumi and Smarandache 2015; Deli 2017; Majumdar et al. 2015; Sahin and Kargn 2019; Zhang et al. 2014). On the basis of fuzzy set theory, Sebastian and Ramakrishnan (2010) introduced multi-fuzzy sets, Atanassov (1986) proposed intuitionistic fuzzy set theory, Shinoj and John (2012)
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