A generalization of Hukuhara difference for interval and fuzzy arithmetic

We propose a generalization of the Hukuhara difference. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is seggested to attempt a generalization of division for real intervals and fuzzy numbers.Fuzzy Arithmetic, Interval Arithmetic, Hukuhara difference, Fuzzy Numbers.

Fuzzy Numbers are the Only Fuzzy Sets that Keep Invertible Operations Invertible

International audienceIn standard arithmetic, if we, e.g., accidentally add a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x + y. In this paper, we prove the following two results: First, a similar possibility to invert (undo) addition holds for fuzzy numbers (although in case of fuzzy numbers, we cannot simply undo addition by subtracting y from the sum).Second, if we add a single fuzzy set that is not a fuzzy number, we lose invertibility.Thus, invertibility requirement leads to a new characterization of the class of all fuzzy numbers