At the time of analyze the Real Number Set R it is essential to associates subsets of R with real numbers . This purpose is served by taking subsets of R in a spacial way as follows .
Open and Closed Intervals :-
An Open Interval of real number is a subset of R defined as
{x/ a<x<b} i.e. it is the set of all real numbers that are greater then a but less then b . This open interval is denoted by
(a ,b) or ]a, b[ .
Thus ; (a, b) = {x/ a<x<b , x is element of R }
and ; y is element of (a, b) => a<y<b .
A closed interval of real number is subset of R defined as
{ x/ a is less then an equal to x is less then and equal to b }
i.e. it is the set of all real numbers that are equal to or greater then a but equal to or less then b . This closed interval is denoted by [a , b] .
Thus;
[a, b]= {x/ a is less then and equal to x is less then and equal to b ,
where x is element of R}
and y is element of [a, b]=> a is less then and equal to y
and y is less then and equal to b .
Clearly open and closed intervals are bounded sets in R . While in case of (a, b) there is no maximum or minimum element , in case of [a, b] the maximum element is b and the minimum element is a .
Here is examples of open , closed ,semi open and semi closed intervals .
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