# Sets/Numbers/Introduction/Section

Mathematical structures like numbers are described as sets. A *set* is a collection of distinct objects which are called the *elements* of the set. By distinct we mean that it is clear which objects are considered to be equal and which are considered to be different. The *containment* of an element to a set is expressed by

the noncontainment by

For every element, exactly one of these possibilities holds. For example, we have
and
.
An important principle for sets is the *principle of extensionality*, i.e. a set is determined by the elements it contains, beyond that it bears no further information. In particular, two sets coincide if they contain the same elements.

The set which does not contain any element is called the *empty set* and is denoted by

A set is called a *subset* of a set if every element from does also belong to . For this relation we write
(some people write
for this). One also says that the *inclusion*
holds. For the number sets, the inclusions

hold. The subset relation
is a statement using for all, as it makes a claim about all elements from . If we want to show
,
then we have to show for an arbitrary element
that also the containment
holds. In order to show this, we are only allowed to use the property
.
For us, sets will be either number sets or sets constructed from such number sets. A set is called *finite* if its elements may be counted by the natural numbers for a certain
.
In this case, the number is called the *number* (or the *cardinality*) of the set.