Fuzzy Set Operations Based on the Theory of Falling Shadows

In this paper, we establish a theoretical approach to define fuzzy set operations based on the theory of falling shadows. The main characteristic of our definition of fuzzy set operations is that it is semantically dependent in the sense that the formula of our definition will vary according to the correlation of the fuzzy sets concerned. We show that the well known formulae of fuzzy set operations such as the max-mm operations, the bounded sum-difference operations, and the probability sum-product operations are consequences of our definition under three different correlations of the fuzzy sets.