Set can be described in three different forms if it contains numerical elements. That are Roaster form, Set builder form and Interval form. While the Roaster and Set Builder notations go with non numerical elements as well, Interval notation can be used only when we are dealing with set of real numbers. Roaster form which lists all the elements of a set is useful only when there a small number of discrete elements. Set Builder form describe the set using the properties of the elements.

Interval notation condenses the real numbers between the boundary values. Let us learn, how to express sets using interval notation and how the descriptions in other notations can be translated to interval notation.

## Definition

Intervals are part of the Real Number line. Interval Notation defines an interval with the boundaries of the interval using parenthesis ( ) or brackets  [ ]. Parenthesis indicates that the end point is not included, while a bracket includes the end point in the interval.

Suppose a and b are real numbers with $a < b$. The four possible notations for intervals with $a$ and $b$ as lower and upper end values along with set builder descriptions are given below.

 Interval Notation Set Builder Description Verbal Description $[a,\ b]$ $\{ x\ |\ a\ \leq\ x\ \leq\ b \}$ This is called a closed interval $a,\ b$.The closed interval includes all realvalues between $a$ and $b$ including the end values $a$ and $b$. $[a,\ b)$ $\{ x\ |\ a\ \leq\ x\ <\ b \}$ This interval includes the lower boundaryvalue $a$, but not the upper end value $b$. $(a,\ b]$ $\{ x\ |\ a\ <\ x\ \leq\ b \}$ This interval includes the upper boundaryvalue $b$, but not the lower end value $a$. $(a,\ b)$ $\{ x\ |\ a\ <\ x\ <\ b \}$ This is called an open interval $a,\ b$. The open interval includes all real values between $a$ and $b$ except the end values$a$ and $b$.

The entire number line or the set of all real numbers is described by the open interval $\left ( -\infty ,+\infty \right )$.
The infinity symbol in the above notation indicates that the set is not bounded both in the positive and negative directions.

## Interval Notation Graph

The graph of an interval can be shown on the number line. Graphs of few intervals are shown below with explanation.

While an open circle or parenthesis is used to represent an open end, a closed circle or a bracket is used to point at the closed end.

 The Graph of the interval $[-2,\infty )$ is shown on the left. The lower boundary -2 is marked with a closed circle or [ to indicate that it is included in the interval. The interval has no upper bound. The Graph of the interval $(-\infty ,5)$ is shown on the right. The upper boundary 5 is marked with an open circle or ")" indicating that it is not included in the interval. The interval has no lower bound. The graph of the finite open interval (-2,5) is shown on the left. Both the end points are marked with open circles or parenthesis, which indicates that the graph represents an open interval.

## How to do Interval Notation

The points to be observed in writing an interval notation are given below.
1) Determine the lower and upper end points.

2) Check whether either or one of the end points is to be included in the interval.

3) Parenthesis is used at the ends to denote the point is not included in the interval.

4) Bracket is used at the ends to denote the point is included.

5) Write the end points as an ordered pair within the closing symbols chosen.

6) Union symbol $U$ is used to express two disjoint intervals together. Example: If the solution of a problem consists of two intervals $(-2,\ 4)$ and $[7,\ 10)$, it is written as $(-2,\ 4)\ U\ [7,\ 10)$.

## Domain an Range in Interval Notation

It is often required to give the domain and range of functions in interval notation.
Example 1:

$f(x)$ = $x^{2} + 1$

The domain of the quadratic function is all real numbers. It is written in interval notation as,

Domain of $f(x)$: $(-\infty, +\infty )$

The minimum function value of $f(x)$ is $1$. It is not bounded above. Hence the range is given in interval notation as

Range of $f(x)$: $\left[ 1, +\infty \right)$
Example 2:

Consider the rational function $h(x)$ = $\frac{x}{x-3}$

Here $h(x)$ is defined for all real values except $x$ = $3$. Hence the domain is expressed as a union of two disjoint intervals.

Domain of $h(x)$: $\left( -\infty, 3 \right)\ \cup\ \left(3, +\infty \right)$

$y$ = $1$ is a horizontal asymptote. Hence $y$ assumes all values except $1$. Thus the range is also expressed as union of intervals.

Range of $h(x)$: $\left( -\infty, 1 \right)\ \cup\ \left ( 1, +\infty \right )$

## Examples

### Solved Example

Question: Describe the set using interval notation.
a) { a | a ≤ 2}    b) { x | x ≠ 4}     c)  { y | -5 ≤ y < 7 }
Solution:

a) a is not bounded below and bounded above by 2 which is also included in the interval.
Hence the interval notation is $(-\infty ,2]$
b) x assumes all real values except 4.  Hence it is represented as $\left ( -\infty ,4 \right )\cup \left ( 4,+\infty \right )$
c) The set can be written as a finite interval, with one open end and another closed as [-5, 7).

2. Write the domain and range of y = sin x in interval notation.
Sin x is defined for all real values of x.
Hence the domain of sin x is $\left ( -\infty ,+\infty \right )$
The minimum and maximum sine values are -1 and 1.
Hence the range of the sine function is [-1, 1].

3. Graph the interval $\left ( -\infty ,-1 \right ]\cup \left ( 6,+\infty \right )$
The disjoint lines going in opposite directions display the union of two intervals.

The interval represented by the left extending line is closed on the right, while the second interval is an open interval.