A class of fuzzy numbers induced by probability density functions and their arithmetic operations

In this paper we are interested in a class of fuzzy numbers which is uniquely identified by their membership functions. The function space, denoted by Xh,p , will be constructed by combining a class of nonlinear mappings h (subjective perception) and a class of probability density functions (PDF) p (objective entity), respectively. Under our assumptions, we prove that there always exists a class of h to fulfill the observed outcome for a given class of p. Especially, we prove that the common triangular number can be interpreted by a function pair (h, p). As an example, we consider a sample function space Xh,p where h is the tangent function and p is chosen as the Gaussian kernel with free variable & mu;. By means of the free variable & mu; (which is also the expectation of p(x; & mu;)), we define the addition, scalar multiplication and subtraction on Xh,p. We claim that, under our definitions, Xh,p has a linear algebra. Some numerical examples are provided to illustrate the proposed approach.& COPY; 2023 Elsevier B.V. All rights reserved.