Element (mathematics)

An element in mathematics is always to be understood in the context of set theory or logic class. The basic relation that when X is an element, and M is a set or class is:

The amount definition of Georg Cantor describes clearly what is meant by a member in connection with a quantity:

This amount considers illustrative of naive set theory is not proved consistent as. Today, therefore, an axiomatic set theory is used, usually the Zermelo -Fraenkel set theory, sometimes a more general class logic.

Examples

Simple examples

Examples of elements can be specified clearly only with respect to the quantity containing them. In mathematics, sets of numbers provide suitable examples:

Specific examples of

In some sub-disciplines of mathematics, certain types of elements occur repeatedly. These special elements are then fixed names.

In group theory are facing specific sets whose elements are interrelated. In such a link then they become a member of the set. There must always be for the sake of defining a group, a particular element, that does not change when linked to any other element of that. This particular element is referred to as a neutral element.

In addition it needs also to each member of the group a counterpart exist due to the definition of the group, which just gives the neutral element link as. This counterpart is called the inverse element ( at a given element).

Within the integers zero is a neutral element with respect to addition. If x is zero added to any number, one again obtains x:

And corresponding to an integer number x -x the inverse element:

Within the real numbers, the number 1 is the neutral element with respect to multiplication. When any real number x multiplied by 1, one again obtains x:

Accordingly, a non-zero real number x is the reciprocal of 1 / X, the inverse element of the multiplication:

More complicated examples

The concept of the element and the amount may also be more complicated. For about a lot contain elements that are themselves sets. Example, one could define a quantity that the amounts already given than did its three elements (: real numbers natural numbers: rational numbers and ):

Would then be ( is the set of natural numbers, a member of the set ).

In fact, to be defined in set-theoretic construction of mathematics in this way the natural numbers formally: