# Sets

In modern mathematics, just about everything rests on the very important concept of the set .

A set is just a collection of elements, or members. For instance, you could have a set of friends:

$\text{F}=$ {Abdul, Gretchen, Hubert, Jabari, Xiomara}

or a set of numbers:

$\text{Y}=\left\{-3.4,12,9999\right\}$

There are two methods of representing a set :

(i) Roster or tabular form

(ii) Set-builder form.

Roster or tabular form: In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

For Example:

$\text{Z}=\text{the}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{set}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{integers}=\left\{\dots ,-3,-2,-1,0,1,2,3,\dots \right\}$

Set-builder form: In the set builder form, all the elements of the set, must possess a single property to become the member of that set.

For Example:

$\text{Z}=\left\{x:x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{an}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{integer}\right\}$

You can read $\text{Z}=\left\{x:x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{an}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{integer}\right\}$ as "The set $\text{Z}$ equals all the values of $x$ such that $x$ is an integer."

$\text{M}=\left\{x\text{\hspace{0.17em}}|\text{\hspace{0.17em}}x>3\right\}$

(This last notation means "all real numbers $x$ such that $x$ is greater than $3$ ." So, for example, $3.1$ is in the set $\text{M}$ , but $2$ is not. The vertical bar | means "such that".)

You can also have a set which has no elements at all. This special set is called the empty set, and we write it with the special symbol $\varnothing$ .

If $x$ is a element of a set $A$ , we write $x\in A$ , and if $x$ is not an element of $A$ we write $x\notin A$ .

So, using the sets defined above,

$-862\in Z$ , since $-862$ is an integer, and

$2.9\notin M$ , since $2.9$ is not greater than $3$ .