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 ( , )-convex fuzzy subset is introduced. The acceptable nontrivial concepts obtained in this manner are the (ε, ε q)-convex fuzzy subset and ( , )-convex fuzzy subset. Second, by the use of the implication operators of fuzzy logic, an R-convex fuzzy subset is proposed. Relations between the ( , )-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.