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

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.