Subsethood, entropy, and cardinality for interval-valued fuzzy sets—An algebraic derivation

• Published in: Fuzzy sets and systems Vol. 158; no. 12; pp. 1384 - 1396
• Main Authors: Vlachos, Ioannis K., Sergiadis, George D.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16.06.2007; Elsevier
• Subjects: Applied sciences; Atanassov's intuitionistic fuzzy sets; Average possible cardinality; Circuit properties; Computer science; control theory; systems; Control theory. Systems; Digital circuits; Electric, optical and optoelectronic circuits; Electronic circuits; Electronics; Entropy-subsethood theorem; Exact sciences and technology; Information, signal and communications theory; Interval-valued fuzzy entropy theorem; Interval-valued fuzzy sets; Mathematical methods; Miscellaneous; Subsethood measures; System theory; Telecommunications and information theory; Theoretical computing; Subsethood measures; Entropy-subsethood theorem; Interval-valued fuzzy sets; Interval-valued fuzzy entropy theorem; Average possible cardinality; Atanassov's intuitionistic fuzzy sets; Fuzzy system; Entropy; Probability distribution; Neural network; Fuzzy set; Axiomatic; Engineering; Information processing; Set theory
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2006.12.018

In this paper a unified formulation of subsethood, entropy, and cardinality for interval-valued fuzzy sets (IVFSs) is presented. An axiomatic skeleton for subsethood measures in the interval-valued fuzzy setting is proposed, in order for subsethood to reduce to an entropy measure. By exploiting the equivalence between the structures of IVFSs and Atanassov's intuitionistic fuzzy sets (A-IFSs), the notion of average possible cardinality is presented and its connection to least and biggest cardinalities, proposed in [E. Szmidt, J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118 (2001) 467–477], is established both algebraically and geometrically. A relation with the cardinality of fuzzy sets (FSs) is also demonstrated. Moreover, the entropy-subsethood and interval-valued fuzzy entropy theorems are stated and algebraically proved, which generalize the work of Kosko [Fuzzy entropy and conditioning, Inform. Sci. 40(2) (1986) 165–174; Fuzziness vs. probability, International Journal of General Systems 17(2–3) (1990) 211–240; Neural Networks and Fuzzy Systems, Prentice-Hall International, Englewood Cliffs, NJ, 1992; Intuitionistic Fuzzy Sets: Theory and Applications, Vol. 35 of Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 1999] for FSs. Finally, connections of the proposed subsethood and entropy measures for IVFSs with corresponding definitions for FSs and A-IFSs are provided.