Sets/Introduction/Section

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

${}x\in M\,,$

the noncontainment by

${}x\notin M\,.$

For every element, exactly one of these possibilities holds.

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 that 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. 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 ${}x\in N$.

Due to the principle of extensionality, we have the following important equality principle for sets, saying that

$M=N{\text{ if and only if }}N\subseteq M{\text{ and }}M\subseteq N$

holds. In mathematical praxis, this means that the equality of two sets is established by proving the two inclusions (in two independent steps). This also has the cognitive advantage that the reasoning gets a direction; it is always clear which conditions can be used and where to go. This principle is analogous to the principle from propositional logic that an equivalence between two statements means both implications, and is best shown by proving the two implications.