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 ${}x$ to a set ${}M$ is expressed by

Georg Cantor (1845-1918) is the creator of set theory.

David Hilbert (1862-1943) has called set theory a paradise, from where mathematicians should never be expelled.

${}x\in M\,,$

the noncontainment by

${}x\notin M\,.$

For every element, exactly one of these possibilities holds. For example, we have ${}{\frac {3}{7}}\notin \mathbb {N}$ and ${}{\frac {3}{7}}\in \mathbb {Q}$. 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

$\emptyset .$

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

${}\mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \,$

hold. The subset relation ${}N\subseteq M$ is a statement using for all, as it makes a claim about all elements from ${}N$. If we want to show ${}N\subseteq M$, then we have to show for an arbitrary element ${}x\in N$ that also the containment ${}x\in M$ holds. In order to show this, we are only allowed to use the property ${}x\in N$. 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 ${}1,2,3,\ldots ,n$ for a certain ${}n\in \mathbb {N}$. In this case, the number ${}n$ is called the number (or the cardinality) of the set.