3.4 Kaucher Interval Arithmetic

Generalized interval arithmetic, also known as Kaucher Interval Arithmetic (), extends the classical interval arithmetic framework () by relaxing the ordering constraint between the bounds. In , the infinimum and supremum are not required to satisfy . This generalization allows for the representation of intervals where the bounds may be reversed, enabling a broader class of mathematical operations and applications.

The set of Kaucher intervals is formally defined as:

Unlike classical intervals, Kaucher intervals can model directed quantities, such as oriented distances or differences, where the sign of the interval carries meaningful information.