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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

Here a certain set of numbers is described by a list of starting elements in the hope that this makes it clear which numbers belong to the set. An important point is that 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 denote a given set. In there are certain elements which fulfil a certain property (a predicate) or not. Hence, for the property we have within the subset consisting of all the elements from which fulfil this property. We write for this subset given by

This only works for such properties for which the statement is well-defined for every . If one introduces such a subset then one gives a name to it which often reflects the name of the property, like

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 of the students in a course, there are many important properties which determine certain subsets, like

The set itself is also given by a property, since