Only Intervals Preserve the Invertibility of Arithmetic Operations

In standard arithmetic, if we, e.g., accidentally added a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x+ y. A similar possibility to invert (undo) addition holds for intervals. In this paper, we show that if we add a single non-interval set, we lose invertibility. Thus, invertibility requirement leads to a new characterization of the class of all intervals. 1 Formulation of the Problem Why Invertible? Many computer operations, including addition x → x + y, are invertible in the sense that if we accidentally added the wrong number y, we can always reconstruct the original value x by simply subtracting y from the result of the addition: (x+ y)− y = x. Case of Partial Knowledge In many real life situations, our knowledge about the actual values x and y is incomplete. This means that we do not know the exact values of x and y, we only know the sets X and Y of possible values of x and y. The most common situation is when the sets X and Y are intervals.