# Element (math)

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

X is an element of M ” or with the help of the element symbol “x ∈ M”.

Georg Cantor's definition of quantities clearly describes what is to be understood by an element in connection with a quantity:

"By a 'set' we mean any combination M of certain well-differentiated objects m of our perception or our thinking (which are called the 'elements' of M ) into a whole."

This descriptive set conception of the naive set theory proved to be not free of contradictions. Today, therefore, an axiomatic set theory is used, mostly the Zermelo-Fraenkel set theory , sometimes also a more general class logic .

## Examples

### Simple examples

Examples of elements can obviously only be given with reference to the quantity they contain. In mathematics, number sets offer suitable examples:

${\ displaystyle 5 \ in \ mathbb {N}}$ 5 is an element of the set of natural numbers
${\ displaystyle {3 \ over 4} \ in \ mathbb {Q}}$ 3/4 is an element of the set of rational numbers
${\ displaystyle {\ sqrt {2}} \ in \ mathbb {R}}$ the square root of 2 is an element of the set of real numbers
${\ displaystyle {\ sqrt {2}} \ notin \ mathbb {Q}}$ the square root of 2 is not an element of the set of rational numbers

### Specific examples

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

In group theory there are special sets whose elements are linked with one another. With such a link , another element of the set is created. For reasons of defining a group, there must always be a special element that does not change it when it is linked to any other element. This particular element is known as the neutral element .

In addition, due to the definition of the group, there must also be a counterpart for each element of the group , which, when linked, results in the neutral element. This counterpart is called the inverse element (of a given element).

Within the whole numbers , zero is a neutral element with regard to addition. If you add zero to any number you get again : ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle {\ begin {matrix} {x + 0 = x} \ end {matrix}}}$ And correspondingly, the number is the inverse element of an integer : ${\ displaystyle x}$ ${\ displaystyle -x}$ ${\ displaystyle {\ begin {matrix} {x + \ left (-x \ right) = 0} \ end {matrix}}}$ Within the real numbers, the number 1 is the neutral element with regard to multiplication . If you multiply any real number by 1, you get again : ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle x \ cdot 1 = x}$ Correspondingly, the reciprocal of a real number other than zero is the inverse element of the multiplication: ${\ displaystyle x}$ ${\ displaystyle {\ tfrac {1} {x}}}$ ${\ displaystyle x \ cdot {1 \ over x} = 1}$ ### More complicated examples

The concept of the element and the set can also be more complex. For example, a set can contain elements that are themselves sets. For example, one could define a set that contains the already mentioned sets ( : natural numbers ,: rational numbers and : real numbers) as its three elements: ${\ displaystyle T}$ ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle T: = \ {\ mathbb {N}, \ mathbb {Q}, \ mathbb {R} \}}$ Then (the set of natural numbers is an element of the set ). ${\ displaystyle \ mathbb {N} \ in T}$ ${\ displaystyle T}$ In fact, in the set-theoretical structure of mathematics, the natural numbers are formally defined in this way:

{\ displaystyle {\ begin {aligned} 0 &: = \ emptyset \\ 1 &: = \ {\ emptyset \} = \ {0 \} \\ 2 &: = \ {\ emptyset, \ {\ emptyset \} \} = \ {0,1 \} \\ 3 &: = \ {\ emptyset, \ {\ emptyset \}, \ {\ emptyset, \ {\ emptyset \} \} \} = \ {0,1,2 \} \ \ & \ \ \ vdots \ end {aligned}}} ## Individual evidence

1. ^ Georg Cantor: Contributions to the foundation of the transfinite set theory. In: Mathematical Annals . Vol. 46, No. 4, , pp. 481-512, doi : 10.1007 / BF02124929 .