Interval Analysis

2Real intervals

2.1Intervals, partial ordering

Real compact intervals are particular subsets of ℝ which form the main objects of this book. They are denoted by square brackets and are defined by

Here, the lower bound inf ([a]) = min([a]) = a and the upper bound sup([a]) = max((a]) = a̅ are real numbers which satisfy a ≤ a̅. In brief, we call [a] an interval, nearly always dropping the specifications real and compact. The set of all intervals [a] is denoted by 𝕀ℝ, that of all intervals [a] contained in some given subset S of ℝ by 𝕀(S). Functions which map 𝕀(S) into 𝕀ℝ are called interval functions. They are denoted, for instance, by [f] : 𝕀(S