![Nullset](https://upload.wikimedia.org/wikipedia/commons/6/6a/Nullset.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the empty set is the unique
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
having no
elements; its size or
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(count of elements in a set) is
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
.
Some
axiomatic set theories
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
ensure that the empty set exists by including an
axiom of empty set
In axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrab ...
, while in other theories, its existence can be deduced. Many possible properties of sets are
vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
for 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".
However,
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
is a distinct notion within the context of
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set.
Notation
Common notations for the empty set include "", "
", and "∅". The latter two symbols were introduced by the
Bourbaki group
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
(specifically
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
) in 1939, inspired by the letter
Ø in the
Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish ance ...
and
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
alphabets. In the past, "0" was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.
The symbol ∅ is available at
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
point U+2205. It can be coded in
HTML
The HyperText Markup Language or HTML is the standard markup language for documents designed to be displayed in a web browser. It can be assisted by technologies such as Cascading Style Sheets (CSS) and scripting languages such as JavaScri ...
as and as . It can be coded in
LaTeX
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
as . The symbol
is coded in LaTeX as .
When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.
Properties
In standard
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
, by the
principle of extensionality, two sets are equal if they have the same elements. 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 empty set has the following properties:
* Its only subset is the empty set itself:
*:
* The
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of the empty set is the set containing only the empty set:
*:
* The number of elements of the empty set (i.e., its
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
) is zero:
*:
For any
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...
set ''A'':
* The empty set is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''A'':
*:
* The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of ''A'' with the empty set is ''A'':
*:
* The
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of ''A'' with the empty set is the empty set:
*:
* The
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of ''A'' and the empty set is the empty set:
*:
For any
property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
''P'':
* For every element of
, the property ''P'' holds (
vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
).
* There is no element of
for which the property ''P'' holds.
Conversely, if for some property ''P'' and some set ''V'', the following two statements hold:
* For every element of ''V'' the property ''P'' holds
* There is no element of ''V'' for which the property ''P'' holds
then
By the definition of
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
, the empty set is a subset of any set ''A''. That is, element ''x'' of
belongs to ''A''. Indeed, if it were not true that every element of
is in ''A'', then there would be at least one element of
that is not present in ''A''. Since there are elements of
at all, there is no element of
that is not in ''A''. Any statement that begins "for every element of
" is not making any substantive claim; it is a
vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
. This is often paraphrased as "everything is true of the elements of the empty set."
In the usual
set-theoretic definition of natural numbers In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Gottlob ...
, zero is modelled by the empty set.
Operations on the empty set
When speaking of the
sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set is zero. The reason for this is that zero is the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
for addition. Similarly, the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the elements of the empty set should be considered to be
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
(see
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
), since one is the identity element for multiplication.
A
derangement
In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.
The number of derangements of ...
is a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of a set without
fixed points. The empty set can be considered a derangement of itself, because it has only one permutation (
), 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
Extended real numbers
Since the empty set has no member when it is considered as a subset of any
ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
, 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 the
real number line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, every real number is both an upper and lower bound for the empty set. When considered as a subset of the
extended reals
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
formed by adding two "numbers" or "points" to the real numbers (namely
negative infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
, denoted
which is defined to be less than every other extended real number, and
positive infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
, denoted
which is defined to be greater than every other extended real number), we have that:
and
That is, the least upper bound (sup or
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
) of the empty set is negative infinity, while the greatest lower bound (inf or
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
) 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
In any
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'', the empty set is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
by definition, as is ''X''. Since the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
of an open set is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and the empty set and ''X'' are complements of each other, the empty set is also closed, making it a
clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
. Moreover, the empty set is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
by the fact that every
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
is compact.
The
closure of the empty set is empty. This is known as "preservation of
nullary
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
unions."
Category theory
If
is a set, then there exists precisely one
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from
to
the
empty function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
. As a result, the empty set is the unique
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
of the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of sets and functions.
The empty set can be turned into a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, called the empty space, in just one way: by defining the empty set to be
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
. This empty topological space is the unique initial object in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
with
continuous maps
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
. In fact, it is a
strict initial object In the mathematical discipline of category theory, a strict initial object is an initial object 0 of a category ''C'' with the property that every morphism in ''C'' with codomain 0 is an isomorphism. In a Cartesian closed category
In category th ...
: only the empty set has a function to the empty set.
Set theory
In the
von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as
. Thus, we have
,
,
, and so on. The von Neumann construction, along with the
axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers,
, such that the
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
of arithmetic are satisfied.
Questioned existence
Axiomatic set theory
In
Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It be ...
, the existence of the empty set is assured by the
axiom of empty set
In axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrab ...
, and its uniqueness follows from the
axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...
. However, the axiom of empty set can be shown redundant in at least two ways:
*Standard
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
implies, merely from the
logical axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, that exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the
axiom of separation
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
.
*Even using
free logic
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter propert ...
(which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the
axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
.
Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an
ontological
In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality.
Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
curiosity, whose meaning and usefulness are debated by philosophers and logicians.
The empty set is not the same thing as ; rather, it is a set with nothing it and a set is always . 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 all
opening moves
Opening may refer to:
* Al-Fatiha, "The Opening", the first chapter of the Qur'an
* The Opening (album), live album by Mal Waldron
* Backgammon opening
* Chess opening
* A title sequence or opening credits
* , a term from contract bridge
* , a te ...
in
chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...
that involve a
king
King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king.
*In the context of prehistory, antiquity and contemporary indigenous peoples, the tit ...
."
The popular
syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
...
: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 "
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
" and the latter to "The set is better than the set
". The first compares elements of sets, while the second compares the sets themselves.
Jonathan Lowe
Edward Jonathan Lowe (; 24 March 1950 – 5 January 2014), usually cited as E. J. Lowe but known personally as Jonathan Lowe, was a British philosopher and academic. He was Professor of Philosophy at Durham University.
Biography
Lowe was bor ...
argues 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, a which has no members. We cannot conjure such an entity into existence by mere stipulation."
George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. in ...
argued that much of what has been heretofore obtained by set theory can just as easily be obtained by
plural quantification
In mathematics and logic, plural quantification is the theory that an individual variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London ...
over individuals, without
reifying sets as singular entities having other entities as members.
George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. in ...
(1984), "To be is to be the value of a variable", ''The Journal of Philosophy
''The Journal of Philosophy'' is a monthly peer-reviewed academic journal on philosophy, founded in 1904 at Columbia University. Its stated purpose is "To publish philosophical articles of current interest and encourage the interchange of ideas, es ...
'' 91: 430–49. Reprinted in 1998, ''Logic, Logic and Logic'' (Richard Jeffrey
Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of pr ...
, and Burgess, J., eds.) Harvard University Press
Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retirem ...
, 54–72.
See also
*
*
*
*
References
Further reading
*
Halmos, Paul, ''
Naive Set Theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. (paperback edition).
*
*
External links
*
{{DEFAULTSORT:Empty Set
Basic concepts in set theory