Empty set

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∅

The empty set is a fundamental concept from set theory . It denotes the set that does not contain any elements. Since sets are characterized by their elements and two sets are equal if and only if they have the same elements (see axiom of extensionality in set theory), there is only one empty set.

The empty set is not to be confused with a zero set , which is a set with the measure zero. Such a set can even contain an infinite number of elements.

Notation and coding

As a symbol for the empty set, the symbol introduced by André Weil and used by Nicolas Bourbaki (a crossed-out circle) has largely prevailed over other notations (such as or ). A typographical variant of this is (a crossed-out narrow oval). Especially in school mathematics, the empty set is also often represented by an empty set brackets: . This sign counteracts a misunderstanding: The empty set is not nothing, but a set that contains nothing. ${\displaystyle \varnothing }$ ${\displaystyle \Lambda }$ ${\displaystyle {\mathit {\Phi }}}$ ${\displaystyle \emptyset }$ ${\displaystyle \left\{\right\}}$

The ∅ is coded as or as in HTML ; in Unicode as and in LaTeX as . Alternatively there is the symbol in LaTeX that is generated by. It should not be confused with the similar-looking diameter symbol ⌀, which is coded as , or the Scandinavian letter Ø ( or ). ∅∅U+2205\varnothing ${\displaystyle \emptyset }$\emptysetU+2300U+00D8U+00F8

Empty set axiom

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

${\displaystyle \exists M\colon \forall X\colon \lnot (X\in M)}$

The uniqueness of the empty set follows from the axiom of extensionality . The existence of the empty set follows with the axiom of exclusion from the existence of any other set. In ZF, which demands the existence of a set in the infinity axiom, the empty set axiom is thus dispensable.

properties

• The empty set is a subset of every set:

  ${\displaystyle \emptyset \subseteq A}$

• Lots of remains in union unchanged with the empty set:

  ${\displaystyle \emptyset \cup A=A}$

• For any set, the average with the empty set is the empty set:

  ${\displaystyle \emptyset \cap A=\emptyset }$

• For any set, the Cartesian product with the empty set is the empty set:

  ${\displaystyle \emptyset \times A=\emptyset }$

• The only subset of the empty set is the empty set:

  ${\displaystyle A\subseteq \emptyset \Rightarrow A=\emptyset }$

• It follows that the power set of the empty set contains exactly one element, namely the empty set itself:

  ${\displaystyle {\mathcal {P}}(\emptyset )=\left\{\emptyset \right\}}$

• For every contradicting statement or property that cannot be fulfilled, the following applies: ${\displaystyle E(x)}$

  ${\displaystyle \emptyset =\left\{x\mid E(x)\right\}}$, e.g. B. ${\displaystyle \emptyset =\left\{x\in \mathbb {Z} \mid x+1=x+2\right\}}$

Thus the empty set is in particular the solution set of an equation or inequality that has no solution.

• Every statement of existence about elements of the empty set, for example

  "There is an x ​​from , so that ..." ${\displaystyle \emptyset }$

is wrong because there is no element that could satisfy the condition.

• Any general statement about elements of the empty set, for example

  "The following applies to all elements of the set ..." ${\displaystyle \emptyset }$

is true because there is no element for which the claim in question could be false.

• Be a set and a figure. Then is the empty set. ${\displaystyle A}$ ${\displaystyle f\colon A\to \emptyset }$ ${\displaystyle A}$
• The empty set is the only basis of the zero vector space .
• The empty set is by definition in any topological space at the same time complete and open .
• Every finite partial cover contains the empty set, so the empty set is compact .
• Likewise, by definition, the empty set is a measurable set in every measure space and has the measure 0.

The empty function

The empty set is in particular an empty set of ordered pairs and thus a mapping . Therefore there is exactly one map for each set ${\displaystyle A}$

${\displaystyle f\colon \emptyset \to A}$,

namely , the so-called empty map or function . You can also put it this way: ${\displaystyle f=\emptyset }$

The empty set is the starting object in the category of sets.

In contrast, there is only one function . ${\displaystyle A=\emptyset }$ ${\displaystyle A\to \emptyset }$

Cardinality of the empty set

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

${\displaystyle \left\vert \emptyset \right\vert =0.}$

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

The empty set is also the only set that is already uniquely determined by its cardinality. (For every other cardinal number the class of the sets of this cardinality is even real .)

literature

• Oliver Deiser: Introduction to set theory . Georg Cantor's set theory and its axiomatization by Ernst Zermelo. 3. Edition. Springer Verlag, Berlin, Heidelberg 2010, ISBN 978-3-642-01444-4 , doi : 10.1007 / 978-3-642-01445-1 .

Web links

Wikibooks: Math for non-freaks: Empty crowd and all class  - learning and teaching materials

Individual evidence

1. ↑ Deiser, p. 31.
2. ^ Willard van Orman Quine : Set Theory And Its Logic . Belknap Press of Harvard University Press, Cambridge, USA 1963, ISBN 0-674-80207-1 , p. 359 (HC) / 380 (PB) (English). - Here: page 19.
   Willard van Orman Quine : Set theory and its logic (=  logic and foundations of mathematics (German translation) . Volume 10 ). Vieweg + Teubner Verlag, 1973, ISBN 3-528-08294-1 , pp. 264 . - Here: page 14.
3. ↑ Akihiro Kanamori: The Empty Set, The Singleton, And The Ordered Pair. In: The Bulletin of Symbolic Logic. Vol. 9, No. 3, Sept. 2003, page 289 ( citing Norbert Wiener ).