Interval

A simple but important kind of set is an interval. Given two real numbers a and b with a < b, the closed interval [a, b] is defined as the set of all real numbers x such that a ≤ x and x ≤ b, or more concisely, a ≤ x ≤ b.

The open interval (a, b) is defined as the set of all real numbers x such that a < x < b. Closed and open intervals are illustrated in Figure 1.1.2.

Figure 1.1.2

For both open and closed intervals, the number a is called the lower endpoint, and b the upper endpoint. The difference between the closed interval [a, b] and the open interval (a, b) is that the endpoints a and b are elements of [a, b] but are not elements of (a, b). When a ≤ x ≤ b, we say that x is between a and b; when a < x < b, we say that x is strictly between a and b.

Three other types of sets are also counted as open intervals: the set (a, ∞) of all real numbers x greater than a; the set (- ∞, b) of all real numbers x less than b, and the whole real line R. The real line R is sometimes denoted by (-∞, ∞). The symbols ∞ and -∞, read "infinity" and "minus infinity," do not stand for numbers; they are only used to indicate an interval with no upper endpoint, or no lower endpoint.

Besides the open and closed intervals, there is one other kind of interval, called a half-open interval. The set of all real numbers x such that a ≤ x < b is a half-open interval denoted by [a, b). The set of all real numbers x such that a ≤ x is also a half-open interval and is written [a, ∞). Here is a table showing the various kinds of intervals.