Open interval as against inclusive interval is a concept often found confusing. In reality there is nothing confusing about an open interval. If we take a set of numbers between two numbers a and b (a < b) and the two end points a and b do not belong to the set i.e. and b are not included in the set, then the interval is called an open interval.
In terms of inequalities we can express open interval as a < x < b. The first inequality a is less than x implies that a is not included, while the second inequality x is less than b implies that b is also not included in the set.Notation for Open Interval
It is common to denote open intervals in mathematics using parenthesis ( ). Thus the open interval discussed above will often be shown as (a, b), though some authors may prefer to follow different conventions.
There can also be intervals that are open on one side and inclusive (or closed) at the other. Such intervals sometimes get called half open (half closed) or more specifically open on the left (right) or inclusive (closed) on the right (left) intervals.Examples of Open and Inclusive Intervals
(1, 2) – is an open interval (non-inclusive) i.e. excluding both the end points 1 and 2 and given by the inequalities
1 < x < 2
[1, 2) – is an interval inclusive on the left but open on the right, i.e. including 1 but excluding 2 and given by the inequalities
1 < = x < 2
(1, 2] – is an interval inclusive on the right but open on the left, i.e. excluding 1 but including 2 and given by the inequalities
1 < x <= 2
[1, 2] – is an inclusive interval given by the inequalities
1 < = x <= 2.
An interesting feature is that all open intervals have the same cardinality, i.e. all numbers in one can be mapped onto the numbers of the other.For example, open interval (1, n) can be mapped on to the open interval (0, 1) by means of the function:
f(x) = `(x-1)/(n-1)``.`Applications of Open Intervals
A conversion of functions with infinite real domain to functions with domain in a finite open interval often makes it easier to comprehend and solve problems in integral calculus and other branches
It is easy to find other functions for providing mappings of any open interval on to any other open interval using the method. This property of open intervals makes them an extremely useful tool in the study of real numbers, functions and calculus.