Fuzzy partitions and transformations

This paper is devoted to an interpretation of fuzzy sets on the basis of indistinguishability. Here we restrict to indistinguishability induced by tolerance bounds ε that lead to identifying numbers whose difference is less than ε. We admit the modification of the ordinary distance by scaling functions. In this way we can establish a one to one correspondence between commonly used fuzzy partitions and sets of crisp points taking the indistinguishability into account. Besides the intuitively appealing interpretation of fuzzy sets we also obtain a method to derive from a fuzzy partition a scaling function that enables us to define a corresponding transformation that simplifies a `sophisticated´ fuzzy partition to a simple one, yielding better utilization of memory and more efficient computation strategies