# What is the difference between set notation and interval notation?

Jun 25, 2018

See below

#### Explanation:

As the question states - it's just a different notation to express the same thing.

When you represent a set with set notation, you look for a characteristic that identifies the elements of your set. For example, if you want to describe the set of all number greater than $2$ and less than $10$, you write

$\setminus \left\{x \setminus \in \setminus m a t h \boldsymbol{R} | 2 < x < 10 \setminus\right\}$

Which you read as "All the real number $x$ ($x \setminus \in \setminus m a t h \boldsymbol{R}$) such that (the symbol "|") $x$ is between $2$ and $10$ ($2 < x < 10$)

On the other hand, if you want to represent the set with interval notation, you need to know the upper and lower bound of the set, or possibly the upper and lower bound of all the intervals that compose the set.

For example, if your set is composed by all the numbers smaller than $5$, or between $10$ and $20$, or greater than $100$, you write the following union of intervals:

$\left(- \setminus \infty , 5\right) \setminus \cup \left(10 , 20\right) \setminus \cup \left(100 , \setminus \infty\right)$

This same set can be written in set notation:

$\setminus \left\{x \setminus \in \setminus m a t h \boldsymbol{R} | x < 5 \text{ or " 10 < x < 20 " or } x > 100 \setminus\right\}$

Finally, note that if the characterization of the set is rather complex, the set notation becomes preferable to the interval one, which would require a great number of intervals in the union. In some other cases, it could be literally impossible to write a set in interval notation, for example is you consider only irrational numbers, you write

$\setminus \left\{x \setminus \in \setminus m a t h \boldsymbol{R} | x \setminus \notin \setminus m a t h \boldsymbol{Q} \setminus\right\}$

but you can't write is as union of intervals.

Jun 25, 2018

See explanation below

#### Explanation:

Imagine we have to express $\left[a , b\right]$ in set notation

$A = \left[a , b\right]$, then $A = \left\{x \in \mathbb{R} / a \le x \le b\right\}$

In this notation we define the characteristics of all $x$ belonging to this set $A$ ....x must be greather or equal to a and simultaneusly samaller or equal to b...

Interval notation is other way to say the same but assuming that [ means the extreme a is IN the interval and ( means extreme $a$ is not.