Empty set

The empty set is a fundamental concept from set theory. It refers to the set that contains no elements. Since quantities are characterized by their elements and two sets are equal if and only if they have the same elements, there is only one empty set.

The empty set is not to be confused with a zero amount, such may even contain an infinite number of elements.

Notation and coding

As a sign of the empty set is the character used by Nicolas Bourbaki ( a slashed circle ) has largely prevailed. A typographic variant thereof is ( a crossed narrow oval ). Especially in school mathematics, the empty set is also often represented by an empty set clip. This character acts against a misunderstanding: The empty set is not nothing but a set that contains nothing.

The ∅ is in HTML or as ∅ as ∅ coded; varnothing as \ in Unicode as U 2205 and in LaTeX. Alternatively, there is the symbol in LaTeX that is generated by emptyset \. You should not confuse it with the similar aussehendem diameter ⌀ characters encoded as U 2300, or the Scandinavian letter Ø (U 00 D8 and U 00 F8).

Empty set axiom

An axiom which requires the existence of an empty set, was first formulated in 1907 by Ernst Zermelo in Zermelo set theory. It was later adopted in the Zermelo -Fraenkel set theory ZF and other axiomatic set teachings. This empty set axiom is verbally: There is a set that contains no elements. The precise logical formula is:

The uniqueness of the empty set follows from the axiom of extensionality. The existence of the empty set follows the axiom of the existence of any other set. In ZF, which calls for the existence of a set in the axiom of infinity, the empty set axiom is thus unnecessary.

Properties

• The empty set is a subset of any amount:
• Any amount will remain unchanged when combined with the empty set:
• For each quantity the average is with the empty set is the empty set:
• For each set is the Cartesian product of the empty set is the empty set:
• The only subset of the empty set is the empty set:
• It follows that the power set of the empty set contains exactly one element, namely the empty set itself:
• For any contradictory statement or not satisfiable property applies: , For example,

• Each existential statement about elements of the empty set, about "It exists from an x such that ... "

• Each All- statement about elements of the empty set, about " For all elements of the set ... "

• For each set there exists a mapping

• Be a quantity and a picture. Then the empty set.
• The empty set is the only basis of the zero vector space.
• The empty set is defined to be completed at the same time in any topological space and open.
• Also, by definition, is the empty set in every measure space is a measurable quantity and has the measure 0

Cardinality of the empty set

The empty set is the only set with cardinality ( cardinality ) is zero:

Therefore, it is also the only representative of the cardinal number 0 and the ordinal 0 In particular, it is a finite set.

The empty set is the only set that is already determined by its cardinality clearly, for any other cardinal number on the other hand, the class of quantities of this cardinality is even real.