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Interval Notation

Interval means a time period that passes in between the occurrence of two events. In mathematics, interval means the numbers between two given numbers. Suppose we have to tell that the weight of the ball can be 8, 9 or 10 units. Then, it can be said that the weight lies in the interval of 8 to 10 units.

Example: Interval between two numbers 3 and 7 is starting from 3 and ends on 7 some numbers in intervals are 3.1, 3.2,…..., 4, 4.1, ...,5, 6,…. 6.9. This is called an open interval, since endpoints are not included. On the other hand, when both end points are also included,  it is called closed interval.
The ending numbers depend upon the given problem with notation.

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An interval notation is used to represent a certain set of continuous data where the elements are not repeated. If we have to say that x can be any number from 3 to 8, we can write it as x $\in$ [3,8]. An interval notation can be broadly classified in three ways:

  1. Open interval: In such notation, the endpoints are not included in the set. It is denoted by endpoints enclosed within small brackets. If it is given x $\in$ (3,8),then it can be understood as 3 < x < 8.
  2. Closed interval: In an interval notation with closed intervals, the end points are also included. Closed interval is represented by  enclosing endpoints inside big brackets. If it is given that x $\in$ [4, 7], it can be understood as $4\leq x \leq 7$.
  3. Half open or half closed: There is another way of writing an interval notation which is said to be half open and half closed. For   example: x $\in$ (2,7], it implies that $2 < x \leq 7$. Similarly, x $\in$ [2,7) implies that $4 \leq x < 7$.

How to Do Interval Notation

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In order to represent interval notation, one needs to follow the steps mentioned below.
Step 1 : When an open end of the interval is given, we use hollow circle (o) to represent it on the number line. When there is a closed end, we denote it by solid circle on number line.

Step 2 : From the first endpoint of given interval, draw a line through second endpoint. If given interval is the union of two intervals, draw both intervals on the same number line.
For example : Have a look at the following interval.

Interval notation infinity

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Suppose we need to represent a variable x such that the value of x can be anything greater than 5. What will be the endpoints? The starting point will be 5 and the end point will taken as infinity. Since infinity is not a particular number, hence the endpoint with infinity cannot be taken as a closed interval, it will always be an open interval. The infinity can be both positive and negative. To represent a number greater than 5, we use the notation (5, $\infty$). To represent any number which is less than or equal to -3, the notation is used will be ($-\infty$, -3].

Usage of interval notation in writing Domain and Range

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The interval notation is used to represent a set of continuous and non-repeating data. A function consists of a dependent and an independent variables. The domain of a function will have set of values that fit in for the independent variable and range is the values that come for the dependent variable. The interval notations can be used to represent the domain and range. If a function $f(x) = x^{2}$. The domain will be of all real numbers. The range will be (0, $\infty $).

Set interval notation

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Set interval notation may be used to represent the values which are not continuous, in the form  of interval notation. Suppose the value of a variable can be written as all the values less than 3 and greater than 5. This set contains all numbers below 3 and above 5, i.e. all the numbers except 3, 4 and 5. How can we write this set in interval notation? It can be written as $(-\infty, 3) \cup (5, \infty )$. The concept of the union of two sets is being used here.

Interval Notation Symbols

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The notation being used for expressing interval notation are discussed below.
Small Braces ( )
The small brackets are used to denote an open interval. When an interval is given as (a, b), it means that it includes all the numbers between a and b, but not a and b themselves.

Big Braces [ ]
The curly brackets are used to represent a closed interval. Suppose an interval is given as [a, b], this would include a, b and all the numbers between a and b.

Union - U
The symbol U is also used as a part of interval notation whenever there are two different intervals are summed up in one. For example I = (a, b) $\cup$ [c, d] implies that interval I represents all the values between a and b excluding a and b along with all the values between c and d including c and d.

Number line
It is used for the graphical representation of an interval. A number line is shown below :

Graphs of Interval Notation

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Let us take a variable ‘x’ and define the interval. If -3 < x < 2 or (-3, 2)

Open Interval Between - 3 to 2

x lies between the interval -3 and 2 which doesn’t included end points -3 and 2 in interval is known as open interval from -3 to 2. On the graph, it is represented by hollow circles.

For  -3 $\leq$ x $\leq$ 2 or [-3, 2]

Close Interval Between - 3 to 2

Here, endpoints -3 and 2 are included, thus it is a closed interval from 3 to 2. It is denoted on graph by solid circles.

For -3 $\leq
$ x < 2

Left Close and Right Open Interval

x lies between the interval -3 and 2 which included end points -3 but doesn’t include 2 in interval is known as left-closed, right-open interval from -3 to 2.

For   -3 < x $\leq$ 2 or (-3, 2]

Left Open and Right Close Intervqal

x lies between the interval -3 and 2 which doesn’t included end points -3 but include 2 in interval is known as left-open, right-closed interval from -3 to 2.

Let us take an example of infinite interval.
If we have x $\geq$ 
2 or [2, $\infty$)

Interval x in greater then 2 which include 2

x lies between 2 (inclusive) and $\infty $ (infinite) which include 2 but the $\infty$ show that it must be any number more then 2. So we can not define maximum limit of x. So we put open braces sign. This is also know as non-ending Interval.


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Let us understand this concept more clearly with the help of following solved examples.

Example 1: Write the value of x in interval notation if x is greater than or equal to 24 and less than 899.

Solution: Here, 24 is included in the interval, while 899 is not. Therefore, we have the following interval notation x $\in$ [24, 899)
Example 2: If x will have a value greater than 8 and less than or equal to 5, then represent it in interval notation.

Solution: Here, x lies between 5 (inclusive) and 8 (non-inclusive), i.e. $x \in (-\infty ,5] \cup (8, \infty)$
Example 3: Represent $x \in (-1 ,5) \cup [9,\infty )$ graphically.

Solution: The given figure will represent the required interval.

Interval Notation Examples
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