intervalialgebra

Interval algebra is a branch of mathematics that deals with the properties and operations of intervals. An interval is a set of real numbers that lies between two endpoints. These endpoints can be included or excluded from the set, leading to different types of intervals such as closed intervals [a, b], open intervals (a, b), half-open intervals [a, b) or (a, b]. Interval algebra extends standard algebraic operations like addition, subtraction, multiplication, and division to these intervals.

The core idea is to define how operations on individual numbers translate to operations on sets of numbers represented by intervals. For example, the sum of two intervals [a, b] and [c, d] is defined as the interval [a+c, b+d]. Similarly, the product of [a, b] and [c, d] is the interval [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]. These operations have applications in various fields where uncertainty or imprecision is present.

Interval arithmetic is used in areas such as computer graphics, numerical analysis, and robotics. It provides a way to rigorously bound the results of calculations when the input values are not known precisely but are known to lie within certain ranges. This is particularly useful for error propagation analysis, where the potential range of an output value is determined by the ranges of its input variables. The theoretical foundations of interval algebra are built upon set theory and real analysis.