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Interval Notation

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In real life we do take a range of numbers to represent a data. For example, my french course is from January to April. We do not say that my french course will be on January, February, March, April. We just take the interval from January to April. In mathematics also many a times a range of numbers does come in use. To represent a set of numbers, the interval notation is used. An interval notation has a start point and an end point, and it does include all the numbers within those points.
 
There are basically three types of interval notations:

1) Open interval

2) Half open interval/Half closed interval

3) Closed interval

Definition

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An interval notation does include all the real numbers within the interval. For example, the interval [1, 2] will include 1.1, 1.2, 2.1 and so on. There are different types of interval notations:
1)  Open interval: An open interval (a, b) will include all real numbers from a to b excluding a and b.
$(a, b)= {x|a<x<b}$.

2)  Closed interval: A closed interval [a, b] will include all real numbers from a to b including a and b. $[a, b]= {x|a\leq x\geq b}$.

3)  Half open interval: An interval [a, b) or (a, b] is known as half open interval. For the interval (a, b] the values greater than a and less than equal to b will be includes. $(a, b]= {x|a< x\geq b}$. Similarly, $[a, b)= {x|a\leq x< b}$.

Domain and range

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Domain of a function defines the acceptable values for the independent variable of the function which will make the function give real output. Range is the set of values that comes for the dependent variable y = f(x).

These values can be represented using the interval notation as well as the set builder notation. Example: for the function y = x - 3 the domain can be given as [3, $\infty $) for the range to be set of non-negative integers.

Inequality

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For some problems to define the set of numbers being used, inequality is used. For example, x will be including all real numbers less than 5 and greater than 10. We need to represent this set in interval notation.
It can be represented as ${x|x\epsilon (-\infty ,5)\cup (10,\infty )}$. Here we can see that either x can be any real number from the negative number line to all real numbers on the positive scale which is less than 5 or it can be any real number greater than 10.

Converter

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The set builder notation is form of writing a set of numbers. A set of numbers given in any type of interval notation can be changed into set builder notation. Suppose an interval (3, 5) is given. It can be written as ${x|3<x< 5}$.

But if this interval is [3, 5] it will be written in the set builder notation as ${x|3\leq x\geq 5}$.

Graph

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The graph of an interval can be represented on a number line. A closed interval [-1, 1] can be represented as given below:
Closed Interval Graph
If the interval is open, (-1, 1) It can be represented as the given figure. We can clearly see that the end points are not being included:
Open Interval Graph
Also we have seen some intervals like [a, $\infty$) which are known as non-ending intervals. When infinity is there only open interval will be used as we definitely cannot include infinity.
 
An interval [4, $\infty$) can be represented as in the given figure:

Interval Notation Graph

Examples

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Example 1: Write the interval [5, 7) in set builder notation.

Solution: The interval [5, 7) will represent all the real numbers between 5 and 7 including 5. It can be written in the set builder notation as {x| 5 $\geq x < 7}.

Example 2: Represent (-$\infty$, 2] graphically.

Solution: (-$\infty$, 2] can be represented graphically on the number line as given here.

Interval Notation

Example 3: For the square root function over the set of real numbers, what will be the domain of the function?

Solution: The square root function will be $y=\sqrt{x}$. The values of x which will be give real values for y can only be the real numbers which are equal to or greater than zero. A negative number will give a imaginary value to y. Hence, interval of domain of the square root function will be $[0,\infty )$.