Dual extension principle and dual fuzzification using lower level sets: representation theorems

Abstract

ABSTRACT The extension principle is always used to fuzzify the crisp functions to obtain the fuzzy functions in which the α-level sets of fuzzy sets are involved in the calculation. In this paper, we shall consider the concepts of upper and lower α-level sets of fuzzy sets. On the other hand, we shall establish the so-called dual extension principle using the concept of dual fuzzy set, which can also be used to fuzzify the crisp function into fuzzy functions. This kind of technique can also be called the dual fuzzification. The benefit of this paper is to provide another technique called dual fuzzification to fuzzify the crisp function into fuzzy functions. The relationship between the fuzzified function using the extension principle and the dual-fuzzified function using the dual extension principle will be established in this paper.