Sets are one of the most fundamental concepts in mathematics. A set is a collection of distinct objects and is considered an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered together they form a single set of size three, written $\left \{ 2,4,6 \right \}$.

Defining a Set

There are two ways of describing, or specifying the members of, a set. One way is through intentional definition, using a rule or semantic description. For example: "$A$ is the set whose members are the first four positive integers."

The second way of describing a set is through extension: listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets: $C = \left \{ 4, 2, 1, 3 \right \}$.

Every element of a set must be unique; no two members may be identical. All set operations preserve this property. The order in which the elements of a set are listed is irrelevant (unlike for a sequence). Therefore:

$\left \{ 6, 11 \right \} = \left \{ 11, 6 \right \} = \left \{ 11, 6, 6, 11 \right \}$

because the extensional specification means merely that each of the elements listed is a member of the set.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as:

$\left \{ 1,2,3, \cdots , 1000 \right \}$

where the ellipsis ($\cdots$) indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as $\left \{ 2,4,6,8, \cdots \right \}$.

Subsets and Supersets

A subset is a set whose every element is also contained in another set. For example, if every member of set $A$ is also a member of set $B$, then $A$ is said to be a subset of $B$. This is written $A \subseteq B$ (also pronounced "$A$ is contained in $B$"). Equivalently, we can say that $B$ is a superset of $A$, which means that $B$ includes $A$, or $B$ contains $A$. This is written $B \supseteq A$.

For example, $\left \{ 1,3 \right \} \subseteq \left \{ 1,2,3,4 \right \}$.

Common Sets

Some of the most commonly referenced sets of numbers are as follows.

The set of natural numbers, also known as "counting numbers," includes all whole numbers starting at 1 and then increasing. The set of natural numbers is represented by the symbol $\mathbb{N}$ and can be denoted as $\mathbb{N}=\left \{ 1,2,3,4, \cdots \right \}$.

The set of real numbers includes every number, negative and decimal included, that exists on the number line. The set of real numbers is represented by the symbol $\mathbb{R}$.

The set of integers includes all whole numbers (positive and negative), including $0$. The set of integers is represented by the symbol $\mathbb{Z}$. (This may seem odd, but it stands for the German term "Zahlen," which means "numbers.") 

The set of rational numbers, denoted by the symbol $\mathbb{Q}$, includes any number that is written as a fraction. The symbol $\mathbb{Q}$ is used because Q represents the word "quotient".

The set of imaginary numbers, denoted by the symbol $\mathbb{I}$, includes all numbers that result in a negative number when squared.

The set of complex numbers, denoted by the symbol $\mathbb{C}$, includes a combination of real and imaginary numbers in the form of $a+bi$ where $a$ and $b$ are real numbers and $i$ is an imaginary number.