Intervals

Interval is a way to describe continuous set of real number by the number that bound them. However, they are not meant to denote a specific point. Rather, they are meant to describe an inequality or system of inequalities.

Let a and b be two distinct real number with a<b then,

1. Open interval

The set of all real number between a and b said to form an open interval from a to b denoted by (a,b). In symbols

           
Geometrically the open interval (a,b) is represented on the real line as

For example= Inequality: -1<x<5
                            Interval    : (-1,5)
In this case, x does not equal -1 and 5. When both of the end point are excluded from the interval, the interval is open interval.

2. Closed interval 

The set of all real number between a and b including the end points and b is said to form a closed interval and is denoted by [a,b].
In symbols 
Geometrically, the closed interval [a,b] is represented on the real line as

For example= inequality: 3≤x≤9
                          interval    : [3,9]
In this case, x could equal 3 or 9 when both of the end point are included in the interval, the interval is a closed interval.

3. Half-open intervals

An interval in which one end point is included are the other end is excluded is called half-open interval.
In symbols, 
Geometrically, [a,b) is represented on the line as

For example= inequality: -3≤x<5
                          interval    : [-3,5]
In this case, x could equal -3 but it cannot equal 5. When one of the end points is included in the interval but the other is not, the the interval is a half-open interval.

Similarly 
Geometrically, (a,b] is represented on the real line as

The interval defined above are called finite intervals. Now we define infinite intervals.

Infinite interval. The set of all real number x such that x>a forms an infinite set and is denoted by (a,∞).
Symbolically  geometrically

If an interval has no lower bound or upper bound then the -∞ and ∞ symbol are used. These symbols are always used with a parentheses bracket, because infinity is not a number that can be included in a set

For example. inequality: x≤7
                          interval   : (-∞,7)
                     
                         inequality: x>3
                         interval    : (3,∞).