. . . Intervals as subsets of R Let a, b ∈ R and a < b.
Then the set of real numbers { y : a < y < b } is called an open interval and is denoted by ( a , b ) . All the points between a and b belong to the open interval ( a, b ) but a, b themselves do not belong to this interval. The interval which contains the end points also is called closed interval and is denoted by [ a, b ]. Thus [ a, b ] = { x : a ≤ x ≤ b } We can also have intervals closed at one end and open at the other, i.e., [ a, b ) = { x : a ≤ x < b } is an open interval from a to b, including a but excluding b.
( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a. These notations provide an alternative way of designating the subsets of set of real numbers. For example , if A = (– , ) and B = [– , ], then A ⊂ B. The set [ , ∞ ) defines the set of non-negative real numbers, while set ( – ∞ , ) defines the set of negative real numbers.
The set ( – ∞ , ∞ ) describes the set of real numbers in relation to a line extending from – ∞ to ∞ . On real number line, various types of intervals described above as subsets of R , are shown in the Fig . . Here, we note that an interval contains infinitely many points.
For example, the set { x : x ∈ R , – < x ≤ }, written in set-builder form, can be written in the form of interval as (– , ] and the interval [– , ) can be written in set- builder form as { x : – ≤ x < }. Fig . MATHEMATICS The