ABSTRACT
The classical set theory is built on the fundamental concept of “set” of which an individual is either a member or not a member. A sharp, crisp, and unambiguous distinction exists between a member and a nonmember for any well-defined “set” of entities in this theory, and there is a very precise and clear boundary to indicate if an entity belongs to the set. In other words, when one asks the question “Is this entity a member of that set?” The answer is either “yes” or “no.” This is true for both the deterministic and the stochastic cases. In probability and statistics, one may ask a question like “What is the probability of this entity being a member of that set?” In this case, although an answer could be like “The probability for this entity to be a member of that set is 90%,” the final outcome (i.e., conclusion) is still either “it is” or “it is not” a member of the set. The chance for one to make a correct prediction as “it is a member of the set” is 90%, which does not mean that it has 90% membership in the set and in the meantime it possesses 10% non-membership. Namely, in the classical set theory, it is not allowed that an element is in a set and not in the set at the same time. Thus, many real-world application problems cannot be described and handled by the classical set theory, including all those involving elements with only partial membership of a set. On the contrary, fuzzy set theory accepts partial memberships, and, therefore, in a sense generalizes the classical set theory to some extent.
