The closed interval [a, b] is the set of real numbers that is represented as
Here, small letters are used to denote the intervals and their endpoints. The left and right endpoints of an interval x will be denoted by x and x, respectively. Thus, x = [ x, x ].
The intervals x and y are said to be equal if they are the same sets. Operationally, this happens if their corresponding endpoints are equal:
We say that x is degenerate if x = x. Such an interval contains a single real number x. In this sense, we may write equations as 0 = [0, 0].
Intersection, Union and Interval Hull
The intersection of two intervals x and y is empty if either у < x or x < y. We denote the empty set by ф and write x n y = ф, indicating that x and y have no points in common. Otherwise, we may define the intersection x n y as the interval
In the latter case, the union of x and y is also an interval
In general, the union of two intervals may not be an interval. However, the interval hull of two intervals, defined by
is always an interval and can be used in interval computations. We have x и у c x и у for any two intervals x and y.