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Neutrosophic Sets and Systems, Vol. 17, 2017
University of New Mexico
On New Measures of Uncertainty for Neutrosophic Sets Pinaki Majumdar Department of Mathematics, M.U.C Women’s College, Burdwan, India-713104, E-Mail: pmajumdar2@rediffmail.com
Abstract: The notion of entropy of single valued neutrosophic sets (SVNS) was first introduced by Majumdar and Samanta in [10]. In this paper some problems with the earlier definition of entropy has been pointed out and a new modified definition of entropy for SVNS has been proposed.
Next four new types of entropy functions were defined with examples. Superiority of this new definition over the earlier definition of entropy has been discussed with proper examples.
Keywords: Single valued neutrosophic sets, Neutrosophic element, Neutrosophic cube, Entropy, Entropy function, Intuitionistic fuzzy sets, Measure of uncertainty. 2010 AMS Classification: 03E72, 03E75, 62C86
1. Introduction. The first successful attempt towards incorporating non-probabilistic uncertainty, i.e. uncertainty which is not caused by randomness of an event, into mathematical modelling was made in 1965 by L. A. Zadeh [20] through his remarkable theory on fuzzy sets (FST). A fuzzy set is a set where each element of the universe belongs to it but with some ‘grade’ or ‘degree of belongingness’ which lies between 0 and 1 and such grades are called membership value of an element in that set. This gradation concept is very well suited for applications involving imprecise data such as natural language processing or in artificial intelligence, handwriting and speech recognition etc. Although Fuzzy set theory is very successful in handling uncertainties arising from vagueness or partial belongingness of an element in a set, it cannot model all sorts of uncertainties prevailing in different real physical situations specially problems involving incomplete information. Further generalization of this fuzzy set was made by K. Atanassov [1] in 1986, which is known as Intuitionistic fuzzy set (IFS). In IFS, instead of one ‘membership grade’, there is also a ‘non-membership grade’ attached with each element. Furthermore there is a restriction that the sum of these two grades is less or equal to unity. In IFS the ‘degree of non-belongingness’ is not independent but it is dependent on the ‘degree of belongingness’. A fuzzy set can be considered as a special case of IFS where the ‘degree of nonbelongingness’ of an element is exactly equal to ‘one
minus the degree of belongingness’. Intuitionistic fuzzy sets definitely have the ability to handle imprecise data of both complete and incomplete in nature. In applications like expert systems, belief systems, information fusion etc., where ‘degree of non-belongingness’ is equally important as ‘degree of belongingness’, intuitionistic fuzzy sets are quite useful. There are of course several other generalizations of Fuzzy as well as Intuitionistic fuzzy sets like L-fuzzy sets and intuitionistic Lfuzzy sets, interval valued fuzzy and intuitionistic fuzzy sets etc that have been developed and applied in solving many practical physical problems [2, 5, 6, 16]. In 1999, a new theory has been introduced by Florentin Smarandache [14] which is known as ‘Neutrosophic logic’. It is a logic in which each proposition is estimated to have a degree of truth (T), a degree of indeterminacy (I) and a degree of falsity (F). A Neutrosophic set is a set where each element of the universe has a degree of truth, indeterminacy and falsity respectively and which lies between [0-, 1+], the non-standard unit interval. Unlike in intuitionistic fuzzy sets, where the incorporated uncertainty is dependent on the degree of belongingness and degree of non belongingness, here the uncertainty present, i.e. the indeterminacy factor, is independent of truth and falsity values. Neutrosophic sets are indeed more general in nature than IFS as there are no constraints between the ‘degree of truth’, ‘degree of indeterminacy’ and ‘degree of falsity’. All these degrees can individually vary within [0-, 1+].
Pinaki Majumdar. On New Measures of Uncertainty for Neutrosophic Sets