Empty set

Inmathematics,theempty setis the uniquesethaving noelements;its size orcardinality(count of elements in a set) iszero.^[1]Someaxiomatic set theoriesensure that the empty set exists by including anaxiom of empty set,while in other theories, its existence can be deduced. Many possible properties of sets arevacuously truefor the empty set.

Any set other than the empty set is called non-empty.

In some textbooks and popularizations, the empty set is referred to as the "null set".^[1]However,null setis a distinct notion within the context ofmeasure theory,in which it describes a set of measure zero (which is not necessarily empty).

Notation[edit]

Common notations for the empty set include "{ }", " $\emptyset$ ",and"∅".The latter two symbols were introduced by theBourbaki group(specificallyAndré Weil) in 1939, inspired by the letterØ(U+00D8ØLATIN CAPITAL LETTER O WITH STROKE) in theDanishandNorwegianAlpha bets.^[2]In the past, "0" (the numeralzero) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.^[3]

The symbol ∅ is available atUnicodepointU+2205∅EMPTY SET.^[4]It can be coded inHTMLas∅and as∅or as∅.It can be coded inLaTeXas\varnothing.The symbol $\emptyset$ is coded in LaTeX as\emptyset.

When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the Alpha betic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.^[5]

Properties[edit]

In standardaxiomatic set theory,by theprinciple of extensionality,two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

The only subset of the empty set is the empty set itself; equivalently, thepower setof the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., itscardinality) is zero. The empty set is the only set with either of these properties.

For anysetA:

The empty set is asubsetofA
TheunionofAwith the empty set isA
TheintersectionofAwith the empty set is the empty set
TheCartesian productofAand the empty set is the empty set

For anypropertyP:

For every element of $\varnothing$ ,the propertyPholds (vacuous truth).
There is no element of $\varnothing$ for which the propertyPholds.

Conversely, if for some propertyPand some setV,the following two statements hold:

For every element ofVthe propertyPholds
There is no element ofVfor which the propertyPholds

then $V=\varnothing.$

By the definition ofsubset,the empty set is a subset of any setA.That is,everyelementxof $\varnothing$ belongs toA.Indeed, if it were not true that every element of $\varnothing$ is inA,then there would be at least one element of $\varnothing$ that is not present inA.Since there arenoelements of $\varnothing$ at all, there is no element of $\varnothing$ that is not inA.Any statement that begins "for every element of $\varnothing$ "is not making any substantive claim; it is avacuous truth.This is often paraphrased as "everything is true of the elements of the empty set."

In the usualset-theoretic definition of natural numbers,zero is modelled by the empty set.

Operations on the empty set[edit]

When speaking of thesumof the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (theempty sum) is zero. The reason for this is that zero is theidentity elementfor addition. Similarly, theproductof the elements of the empty set (theempty product) should be considered to beone,since one is the identity element for multiplication.^[6]

Aderangementis apermutationof a set withoutfixed points.The empty set can be considered a derangement of itself, because it has only one permutation ( $0!=1$ ), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

In other areas of mathematics[edit]

Extended real numbers[edit]

Since the empty set has no member when it is considered as a subset of anyordered set,every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by thereal number line,every real number is both an upper and lower bound for the empty set.^[7]When considered as a subset of theextended realsformed by adding two "numbers" or "points" to the real numbers (namelynegative infinity,denoted $-\infty \!\,,$ which is defined to be less than every other extended real number, andpositive infinity,denoted $+\infty \!\,,$ which is defined to be greater than every other extended real number), we have that: $\sup \varnothing =\min(\{-\infty,+\infty \}\cup \mathbb {R} )=-\infty,$ and $\inf \varnothing =\max(\{-\infty,+\infty \}\cup \mathbb {R} )=+\infty.$

That is, the least upper bound (sup orsupremum) of the empty set is negative infinity, while the greatest lower bound (inf orinfimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

Topology[edit]

In anytopological spaceX,the empty set isopenby definition, as isX.Since thecomplementof an open set isclosedand the empty set andXare complements of each other, the empty set is also closed, making it aclopen set.Moreover, the empty set iscompactby the fact that everyfinite setis compact.

Theclosureof the empty set is empty. This is known as "preservation ofnullary unions."

Category theory[edit]

If $A$ is a set, then there exists precisely onefunction $f$ from $\varnothing$ to $A,$ theempty function.As a result, the empty set is the uniqueinitial objectof thecategoryof sets and functions.

The empty set can be turned into atopological space,called the empty space, in just one way: by defining the empty set to beopen.This empty topological space is the unique initial object in thecategory of topological spaceswithcontinuous maps.In fact, it is astrict initial object:only the empty set has a function to the empty set.

Set theory[edit]

In thevon Neumann construction of the ordinals,0 is defined as the empty set, and the successor of an ordinal is defined as $S(\ Alpha )=\ Alpha \cup \{\ Alpha \}$ .Thus, we have $0=\varnothing$ , $1=0\cup \{0\}=\{\varnothing \}$ , $2=1\cup \{1\}=\{\varnothing,\{\varnothing \}\}$ ,and so on. The von Neumann construction, along with theaxiom of infinity,which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, $\mathbb {N} _{0}$ ,such that thePeano axiomsof arithmetic are satisfied.

Questioned existence[edit]

Historical issues[edit]

In the context of sets of real numbers, Cantor used $P\equiv O$ to denote " $P$ contains no single point ". This $\equiv O$ notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed $O$ as an existent set on its own, or if Cantor merely used $\equiv O$ as an emptiness predicate. Zermelo accepted $O$ itself as a set, but considered it an "improper set".^[8]

Axiomatic set theory[edit]

InZermelo set theory,the existence of the empty set is assured by theaxiom of empty set,and its uniqueness follows from theaxiom of extensionality.However, the axiom of empty set can be shown redundant in at least two ways:

Standardfirst-order logicimplies, merely from thelogical axioms,thatsomethingexists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from theaxiom of separation.
Even usingfree logic(which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely theaxiom of infinity.

Philosophical issues[edit]

While the empty set is a standard and widely accepted mathematical concept, it remains anontologicalcuriosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing asnothing;rather, it is a set with nothinginsideit and a set is alwayssomething.This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of allopening movesinchessthat involve aking."^[9]

The popularsyllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is $\varnothing$ "and the latter to" The set {ham sandwich} is better than the set $\varnothing$ ".The first compares elements of sets, while the second compares the sets themselves.^[9]

Jonathan Loweargues that while the empty set

was undoubtedly an important landmark in the history of mathematics,… we should not assume that its utility in calculation is dependent upon its actually denoting some object.

it is also the case that:

"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, asetwhich has no members. We cannot conjure such an entity into existence by mere stipulation. "^[10]

George Boolosargued that much of what has been heretofore obtained by set theory can just as easily be obtained byplural quantificationover individuals, withoutreifyingsets as singular entities having other entities as members.^[11]

References[edit]

^^a ^bWeisstein, Eric W."Empty Set".mathworld.wolfram.Retrieved2020-08-11.
^"Earliest Uses of Symbols of Set Theory and Logic".
^Rudin, Walter (1976).Principles of Mathematical Analysis(3rd ed.). McGraw-Hill. p. 300.ISBN 007054235X.
^"Unicode Standard 5.2"(PDF).
^e.g. Nina Grønnum (2005, 2013)Fonetik og Fonologi: Almen og dansk.Akademisk forlag, Copenhagen.
^David M. Bloom (1979).Linear Algebra and Geometry.pp.45.ISBN 0521293243.
^Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008).Elementary Real Analysis,2nd edition, p. 9.
^A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair",p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
^^a ^bD. J. Darling (2004).The Universal Book of Mathematics.John Wiley and Sons.p. 106.ISBN 0-471-27047-4.
^E. J. Lowe (2005).Locke.Routledge.p. 87.
^George Boolos(1984), "To be is to be the value of a variable",The Journal of Philosophy91: 430–49. Reprinted in 1998,Logic, Logic and Logic(Richard Jeffrey,and Burgess, J., eds.)Harvard University Press,54–72.

External links[edit]

Weisstein, Eric W."Empty Set".MathWorld.

[:1-1] Weisstein, Eric W."Empty Set".mathworld.wolfram.Retrieved2020-08-11.

[2] "Earliest Uses of Symbols of Set Theory and Logic".

[3] Rudin, Walter (1976).Principles of Mathematical Analysis(3rd ed.). McGraw-Hill. p. 300.ISBN 007054235X.

[4] "Unicode Standard 5.2"(PDF).

[5] .g. Nina Grønnum (2005, 2013)Fonetik og Fonologi: Almen og dansk.Akademisk forlag, Copenhagen.

[6] David M. Bloom (1979).Linear Algebra and Geometry.pp.45.ISBN 0521293243.

[7] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008).Elementary Real Analysis,2nd edition, p. 9.

[8] A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair",p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.

[Darling-9] D. J. Darling (2004).The Universal Book of Mathematics.John Wiley and Sons.p. 106.ISBN 0-471-27047-4.

[Lowe-10] E. J. Lowe (2005).Locke.Routledge.p. 87.

[11] George Boolos(1984), "To be is to be the value of a variable",The Journal of Philosophy91: 430–49. Reprinted in 1998,Logic, Logic and Logic(Richard Jeffrey,and Burgess, J., eds.)Harvard University Press,54–72.

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v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement(i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number(large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Settypes	Amorphous Countable Empty Finite(hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite(Dedekind-infinite) Recursive Singleton Subset·Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo