INTERVALS

Interval is defined as the set or group of real numbers between given two numbers. 

There are different notations to express the interval. They are-

1. [ ] – It is a square bracket and is used when both the endpoints are included in the set.
2. ( ) – It is a round bracket and is used when the endpoints are excluded in the set.
3. ( ] – It is a semi-open bracket and is used when the left endpoint is excluded and the right endpoint is included in the set. 
4. [ ) – It is a semi-open bracket and is used when the left endpoint is included and the right endpoint is excluded in the set.

OPEN AND CLOSED INTERVALS – 

Suppose a, b ϵ R and a < b. So, the set of real numbers y:a<y<b is called an open interval and can be represented by ( a, b ). All the points between a and b belong to open intervals (a, b) but a, b themselves do not belong to this interval. 

The interval which contains the endpoints is called the closed interval and is denoted by [ a, b].

Hence, [ a, b ] = {x :a≤x ≤b }

There are also the intervals closed at one end and open at the other. In other words, 

[ a, b)= x:a≤x<b   is an open interval from a to b, including a but excluding b.

 ( a, b]={ x:a<x≤b } is an open interval from a to b including b but excluding a. 

These notations provide an alternative way of designating the subsets of a set of real numbers. 

For example- if A= (-3, 5) and B= [-7,9]. Here,  A ϲ B. 

The set [ 0, ꝏ) defines the set of non-negative real numbers. The set ( -ꝏ, 0) defines the set of negative real numbers.  The set ( -ꝏ, ꝏ) describes the set of real numbers in relation to a line extending from -ꝏ to ꝏ. 

On a real number line, various types of intervals are described above as subsets of R. 

To explain the above-discussed intervals, let’s take the example-

The set { x:x ϵ R, -5<x≤7 }, written in set- builder form, can be written in the form of an interval as    (-5, 7] and the interval [-3, 5) can be written in a set- builder form as {x :-3 ≤x<5}.

The number (b-a ) is called the length of any of the intervals ( a, b), [a, b], [a, b) or (a, b]. 

NUMBER LINE-

To represent on Number Line, you have to draw thick lines to show the values which are included and if you want to include the end value, then filled- in a circle or an open circle if you don’t want to include the value.

INFINITE INTERVALS-

Suppose A = { 4,5,6,7}     B = { a, b, c, d, e, f}  and  

C= { children living presently in different parts of the world}

Here, we can observe that A has 4 elements and B has 6 elements. In C, we do not know the number of elements but it would be some natural number. 

By a number of elements of a set S, it is meant that the number of distinct elements of the set  & is represented by n(S). If n(S) is a natural number, then S is called a non-empty finite set.

Now, let us take the set of natural numbers. The number of elements in this set is not finite as there are infinite numbers of natural numbers. So, the set of natural numbers is an infinite set.  

A set that has a definite number of elements or is empty is called a finite set otherwise it is called an infinite set. For example-

1. Let A be the set of days of the week. So, A is finite.
2. Let B be the set of points on a line. So, B is infinite. 

 All infinite sets cannot be represented in the roster form. For example- the set of real numbers cannot be represented in this form because the elements of this set do not follow any particular pattern. 

CONCLUSION –

As discussed, the interval is the set of real numbers between two given groups. The interval which contains the endpoints is called the closed interval. The intervals which do not contain the endpoints are called open intervals. There are also the intervals closed at one end and open at the other. Talking about representing on Number Line, you have to draw thick lines to show the values which are included and if you want to include the end value, then filled- in a circle or an open circle if you don’t want to include the value.