Intervals

The real numbers can be represented geometrically as points on a number line called the real line.

The symbol R denotes either the real number system or the real line. A subset of the real line is called an interval if it contains at least two numbers and contains all the real numbers lying between any two of its elements. For example,

  • the set of all real numbers x such that x > 6
  • the set of all real numbers x such that −2 ≤ x ≤ 5
  • the set of all real numbers x such that x < 5

The set of all natural numbers is not an interval. Between any two rational numbers there are infinitely many real numbers which are not included in the given set. Hence, the set of natural numbers is not an interval. Similarly the set of all non zero real numbers is also not an interval. Here the real number 0 is absent.

Geometrically, intervals correspond to rays and line segments on the real line. The intervals corresponding to line segments are finite intervals and intervals corresponding to rays and the real line are infinite intervals. Here finite interval does not mean that the interval contains only a finite number of real numbers.

A finite interval is said to be closed if it contains both of its end points and open if it contains neither of its end points. To denote the closed set, the square bracket [ ] is used and the paranthesis ( ) is used to indicate open set.

Type of Intervals