SETS AND INTERVAL

We know the sets and their types and also know the their operations , let's know the intervals.

Intervals :

1.    Open Interval :

             Let a ,b ∈ R  and a < b then the set  { x/x ∈ R , a < x < b} is called open  interval and is denoted by        (a ,b)  .All the number between a and b are belongg to this open interval (a ,b) but the number a, b are dose't belong in this interval . 
                      eg ., (3, 8) = {x | 3 < x < 8 }

                

2.   Closed Interval :

              Let a ,b ∈ R  and a < b then the set { x/x ∈ R , a ≤ x ≤ b} is called closed interval  and is denoted by [a, b]. All the number between a and b are belong in this interval [a, b] and also a and b itself are belong in this interval.
     eg.,   [1, 11] = {x | 1 ≤ x ≤ 11 }
 

3.  Semi - Closed Interval :

          Let a ,b ∈ R  and  the set { x/x ∈ R , a ≤ x < b} is called semi - closed interval  and is denoted by [a, b). All the number between a and b are belong in this interval but b dose not belong in th interval
 [a, b) i.e.  a ∈ [a, b)  but b ∉ [a, b) .



4.   Semi - Open Interval:

           Let a ,b ∈ R  and the set { x/x ∈ R , a < x ≤ b} is called semi-open interval and is denoted by       (a, b]. All the number between a and b are belong in this interval but a dose't belong inthis interval .  i.e        b ∈ (a, b]  but a ∉ (a, b] .


5.   i)  The set of all real numbers greater than a i.e. (a, (-∞ , b) = { x/x ∈ R , x < b}) = { x/x ∈ R , x > a}

  
ii) The set of allreal numbers greater then or equal to a  i.e  [a, ∞) = { x/x ∈ R , x ≥ a}     


6.  i) The set all real numbers less then b i.e. (-∞ , b) = { x/x ∈ R , x < b}


 ii) The set of alll real number less then or equal to b i.e (-∞ , b] = { x/x ∈ R , x ≤ b}    



7.     The set of all real numbers R is (-∞ , ∞) = {x /x ∈ R , -∞ < x < ∞}
        

Infinite intervals :  are those that do not have an endpoint in either the positive or negative direction, or both. The interval extends forever in that direction. Infinite intervals are summarized in the table below.


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