Friday, 3 January 2014

Intervals in R

 

At the time of analyze  the Real Number Set   R   it is essential to associates subsets of     with real numbers . This purpose is served by taking subsets of    in a spacial way as follows .

Open and Closed Intervals :-

An Open Interval of real number is  a subset of   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    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   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 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    .

Here is examples of open , closed ,semi open and semi closed intervals .


 \begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R\,|\,a<x<b\}, \\{}
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R\,|\,a\le x<b\}, \\{}
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R\,|\,a<x\le b\}, \\{}
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R\,|\,a\le x\le b\}.
\end{align}


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