The definition of convex fuzzy subset
Following the seminal work of Zadeh on the definition of a convex fuzzy subset, three new kinds of definitions of convex fuzzy subsets are proposed. First, by the use of the relations between fuzzy points and fuzzy sets, the definition of a ((beta) over bar,(alpha) over bar)-convex fuzzy subset is introduced. The acceptable nontrivial concepts obtained in this manner are the (is an element of, is an element of Vq)-convex fuzzy subset and ((is an element of) over bar(is an element of) over bar V (q) over bar )-convex fuzzy subset. Second, by the use of the implication operators of fuzzy logic, an R-convex fuzzy subset is proposed. Relations between the ((beta) over bar,(alpha) over bar -convex fuzzy subset and R-convex fuzzy subset are discussed. Finally, based on the concept of falling shadow, a theoretical approach of convex fuzzy subset is established and the convex fuzzy subset based on t-norm is proposed. (C) 2004 Elsevier Ltd. All rights reserved.
convex fuzzy subset fuzzy point implication operator r-convex fuzzy subset (lambda,mu-convex fuzzy subset t-convex fuzzy subset