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The set of fuzzy threshold functions is defined to be a fuzzy set over the set of functions. All threshold functions have full memberships in this fuzzy set. Defines and investigates a distance measure between a non‐linearly separable function and the set of all threshold functions. Defines an explicit expression for the membership function of a fuzzy threshold function through the use of this distance measure and finds three upper bounds for this measure. Presents a general method to compute the distance, an algorithm to generate the representation automatically, and a procedure to determine the proper weights and thresholds automatically. Presents the relationships among threshold gate networks, artificial neural networks and fuzzy neural networks. The results may have useful applications in logic design, pattern recognition, fuzzy logic, multi‐objective fuzzy optimization and related areas.

The grade of membership function for fuzzy monotone functions is defined and investigated. An algorithm for finding the membership value is presented. A minterm even function and minterm odd function are defined and studied. It is found that these two functions are the two most alternation functions. The relationships with threshold functions are also presented. In addition, three ways to implement a fuzzy monotone increasing function are investigated. Applications to function representation, data compression and error detection are illustrated. The results have useful applications in fuzzy logic, expert systems, fuzzy expert systems, and also management of uncertainty.