Factors Space, A New Frontier to Fuzzy Sets Theory

Professor L. A. Zadeh is a special inspiring leader in cybernetics and machine intelligence. As a tribute to his pioneer work in fuzzy sets theory, this paper follows his treatment in fuzzy membership functions and operations and introduces the factors space theory which further explains the relationships between them. The factors space theory calculates the membership degree of a human concept by first surveying the sampling interval statistics around a concept, and then introducing the vision of 'falling shadow' to convert the covering probability of a random interval in the power set of the concept domain to membership degree. This paper introduces the factorial probability and transfers typical probability density to typical falling membership curves. Based on this idea, a subjective scoring system used on fuzzy evaluation and decision-making is suggested. Facing the question on the selection of fuzzy operations, the paper indicates that the source is in the joint distribution of random sets and a selection proposition is given. Factors space opens a new frontier for the developments of fuzzy sets theory.