×

Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. (English) Zbl 0872.94061

Let \(X\) be a nonempty fixed set (universe). An intuitionistic fuzzy set (IFS) \(A\) is an object having the form \(A=\{\langle x,\mu_A(x),\gamma_A(x)\rangle\): \(x\in X\}\) where the functions \(\mu_A:X\to [0,1]\) and \(\gamma_A:X\to [0,1]\) denote the degrees of membership and of non-membership of each element \(x\in X\) to the set \(A\), resp., and \(0\leq\mu_A(x)+ \gamma_A(x)\leq 1\) for each \(x\in X\) [K. T. Atanassov, Fuzzy Sets Syst. 20, 87-96 (1986; Zbl 0631.03040)]. Let \(IFSs(x)\) and \(FSs(x)\) be the sets of all \(IFSs\) and of all fuzzy sets on \(X\), resp. a real-valued function \(I:IFSs(X)\to\mathbb{R}^+\) is called an intuitionistic fuzzy entropy on \(IFSs(X)\), if \(I\) has the following properties:
1) \(I(A)=0\) iff \(A\in FSs(X)\);
2) \(I(A)= \text{card}(X)\) iff for all \(x\in X\): \(\mu_A(x)= \gamma_A(x)=0\);
3) for all \(A\in IFSs(X)\): \(I(A)= I(\overline{A})\);
4) for all \(A,B\in IFSs(X)\): if \(A\preccurlyeq B\) then \(I(A)\geq I(B)\),
where \(A=\{\langle x,\gamma_A(x), \mu_A(x)\rangle\): \(x\in X\}\). A theorem characterizing the entropy is proved. Some examples of the form of the function \(I\) are given and the basic properties of the intuitionistic fuzzy entropy are studied.

MSC:

94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
03E72 Theory of fuzzy sets, etc.

Citations:

Zbl 0631.03040
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986) · Zbl 0631.03040
[2] Atanassov, K., Two operators on intuitionistic fuzzy sets, (Comptes rendus de l’Académi bulgare des Sciences. Comptes rendus de l’Académi bulgare des Sciences, Tome, 41 (1988)), 35-38, 5 · Zbl 0651.03040
[3] Atanassov, K.; Gargov, G., Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31, 343-349 (1989) · Zbl 0674.03017
[4] Burillo, P.; Bustince, H., Estructuras algebraicas en conjuntos IFS, (II Congreso Nacional de Lógica y Tecnología Fuzzy (1992), Boadilla del Monte: Boadilla del Monte Madrid, Spain), 135-147
[5] De Luca, A.; Termini, S., A definition of nonprobabilistic entropy in the setting of fuzzy theory, J. General Systems, 5, 301-312 (1972) · Zbl 0239.94028
[6] Kaufmann, A., Introduction to the Theory of Fuzzy Subsets (1975), Academic Press: Academic Press New York · Zbl 0332.02063
[7] Loo, S. G., Measures of fuzziness, Cybernetica, 20, 201-210 (1977) · Zbl 0377.04003
[8] Sambuc, R., Functions φ-flous. Aplication a l’aide au diagnostic en pathologie thyrodene,, (Thèse (1975), Universite de Marseille)
[9] Trillas, E.; Riera, T., Entropies in finite fuzzy sets, Inform. Sci., 15, 159-168 (1978) · Zbl 0436.94012
[10] Yager, R. R., On measure of fuzziness and fuzzy complements, Internat. J. General Systems, 8, 169-180 (1982)
[11] Zadeh, L. A., Fuzzy sets and systems, (Proceedings of the Symposium on Systems Theory (1965), Polytechnic Institute of Brooklyn: Polytechnic Institute of Brooklyn NY), 29-37 · Zbl 0263.02028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.