Sets/Descriptions/Introduction/Section

There are several ways to describe a set. The easiest one is to just list the elements of the set, here the order of the listing is not important. For finite sets, this is possible; however, for infinite sets, one has to describe a "construction rule“ for the elements.

The most important set given by an infinite listing is the set of natural numbers

${}\mathbb {N} =\{0,1,2,3,\ldots \}\,.$

Here, a certain set of numbers is described by the first elements in the hope that this indicates how the listing goes on and which numbers belong to the set. An important point is that ${}\mathbb {N}$ is not a set of certain digits, but the set of values represented by these digits or sequences of digits. For a natural number, there are many possibilities to represent it, the decimal expansion is just one of them.

We discuss now the description of sets by properties. Let a set ${}M$ and a certain property ${}E$ (a predicate) be given, such that the property can be applied to the elements of ${}M$. Hence, for the property ${}E$, we have in ${}M$ the subset consisting of all the elements from ${}M$ which fulfil this property. We write for this subset given by ${}E$

${}{\left\{x\in M\mid E(x)\right\}}={\left\{x\in M\mid x{\text{ fulfils property }}E\right\}}\,.$

This only works for such properties for which the statement ${}E(x)$ is well-defined for every ${}x\in M$. If one introduces such a subset, then one gives a name to it, which often reflects the name of the property, like

${}E={\left\{x\in \mathbb {N} \mid x{\text{ is even}}\right\}}\,,$

${}O={\left\{x\in \mathbb {N} \mid x{\text{ is odd}}\right\}}\,,$

${}S={\left\{x\in \mathbb {N} \mid x{\text{ is a square number}}\right\}}\,,$

${}{\mathbb {P} }={\left\{x\in \mathbb {N} \mid x{\text{ is a prime number}}\right\}}\,.$

For the sets occurring in mathematics, a multitude of mathematical properties is relevant, and, therefore, there is a multitude of relevant subsets. But also in the sets of everyday life like the set ${}C$ of the students in a course, there are many important properties which determine certain subsets, like

${}O={\left\{x\in C\mid x{\text{ lives in Osnabr}}{\ddot {\rm {u}}}{\text{ck}}\right\}}\,,$

${}P={\left\{x\in C\mid x{\text{ studies physics}}\right\}}\,,$

${}D={\left\{x\in C\mid x{\text{ has birthday in December}}\right\}}\,.$

The set ${}C$ itself is also given by a property, since

${}C={\left\{x\mid x{\text{ is a student in the course}}\right\}}\,.$