An interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators. I: Case of type 1 fuzzy sets (Q1311799)

Without quoting the relevant source [\textit{I. R. Goodman} and \textit{H. T. Nguyen}, Uncertainty models for knowledge-based systems (North-Holland 1985; Zbl 0576.94001)], this paper interprets once more the known fact that the membership function of a fuzzy set \(A\) can be thought as a one- point-coverage-function of an associated random set (in the language of the above paper: as the relative weight of the number of so-called affirmative sets w.r.t. a so-called ground set) and that different intersection or union operators (usually represented by different \(t\)- (co)norms) have a random-set-interpretation (at least between min (max) and bold intersection (union)) which can be expressed by positive or negative correlation between the associated random sets.