In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and more specifically
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the
empty set is the unique set having no
elements; its size or
cardinality (count of elements in a set) is
zero0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. Some axiomatic set theories assure that the empty set exists by including an
axiom of empty setIn axiomatic set theory, the axiom of empty set is an axiom of Zermelo–Fraenkel set theory, the fragment thereof Burgess calls ST, and Kripke–Platek set theory.- Formal statement :...
; in other theories, its existence can be deduced. Many possible properties of sets are
triviallyIn mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
true for the empty set.
Null setIn mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
was once a common synonym for "empty set", but is now a technical term in measure theory.
Notation
Common notations for the empty set include "{}," "
", and "
". The latter two symbols were introduced by the Bourbaki group (specifically
André WeilAndré Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
) in 1939, inspired by the letter
ØØ — minuscule: "ø", is a vowel and a letter used in the Danish, Faroese, Norwegian and Southern Sami languages.It's mostly used as a representation of mid front rounded vowels, such as ø œ, except for Southern Sami where it's used as an [oe] diphtong.The name of this letter is the same as the sound...
in the
Danish and Norwegian alphabetThe Danish and Norwegian alphabet is based upon the Latin alphabet and has consisted of the following 29 letters since 1917 and 1955 , although Danish did not officially recognize the W as a separate letter until 1980....
(and not related in any way to the Greek letter
ΦPhi , pronounced or sometimes in English, and in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents , a voiceless labiodental fricative. In Ancient Greek it represented , an aspirated voiceless bilabial plosive...
). Other notations for the empty set include "Λ" and "0"
The empty-set symbol is found at
UnicodeUnicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
point U+2205. In
TeXTeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....
, it is coded as \emptyset or \varnothing.
Properties
By the
principle of extensionalityIn 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.- Formal statement :...
, two sets are equal if they have the same elements; therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set".
The mathematical symbols employed below are explained
hereThis is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
.
For any set
A:
- The empty set is a subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of A:
- The union
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of A with the empty set is A:
- The intersection
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of A with the empty set is the empty set:
- The Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of A and the empty set is empty:
The empty set has the following properties:
- Its only subset is the empty set itself:
- The power set of the empty set is a set containing only the empty set:
- Its number of elements (that is, its cardinality) is zero:
The connection between the empty set and zero goes further, however: in the standard
set-theoretic definition of natural numbersSeveral ways have been proposed to define the natural numbers using set theory.- The contemporary standard :In standard, Zermelo-Fraenkel set theory the natural numbers...
, we use sets to
modelIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
the natural numbers. In this context, zero is modelled by the empty set.
For any
propertyIn modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...
:
- For every element of
the property holds (vacuous truthA vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...
);
- There is no element of
for which the property holds.
Conversely, if for some property and some set
V, the following two statements hold:
- For every element of V the property holds;
- There is no element of V for which the property holds,
- then
.
By the definition of
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
, the empty set is a subset of any set
A, as
every element
x of
belongs to
A. If it is not true that every element of
is in
A, there must be at least one element of
that is not present in
A. Since there are
no elements of
at all, there is no element of
that is not in
A. Hence every element of
is in
A, and
is a subset of
A. Any statement that begins "for every element of
" is not making any substantive claim; it is a
vacuous truthA vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...
. This is often paraphrased as "everything is true of the elements of the empty set."
Operations on the empty set
Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the
sumSUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of the elements of the empty set is zero, but the
productMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
of the elements of the empty set is
one (see
empty productIn mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...
). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the
identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
for addition, and one is the identity element for multiplication.
Extended real numbers
Since the empty set has no members, when it is considered as a subset of any
ordered setIn order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...
, then 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, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted
which is defined to be less than every other extended real number, and
positive infinityPositive Infinity are a collaborative group of musicians from Miami, Florida who work with various musical genres. While they are a full featured studio band, Positive Infinity are generally considered to be the solo project of Living Corban's former vocalist and guitarist, Jonathan Roberts...
, denoted
which is defined to be greater than every other extended real number, then:
and
That is, the least upper bound (sup or
supremumIn mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
) of the empty set is negative infinity, while the greatest lower bound (inf or
infimumIn mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
) 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 minimum and infimum.
Topology
Considered as a subset of the real number line (or more generally any
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
), the empty set is both
closedIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and
openThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
; it is an example of a
"clopen" setIn topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
. All its
boundary pointsIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
(of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an open neighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the fact that every
finite set is compact.
The
closureIn mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
of the empty set is empty. This is known as "preservation of nullary
unionsIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
."
Category theory
If
A is a set, then there exists precisely one
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f from {} to
A, the
empty functionIn mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty functionf_A: \varnothing \rightarrow A....
. As a result, the empty set is the unique
initial objectIn category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
of the
categoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of sets and functions.
The empty set can be turned into a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, called the empty space, in just one way: by defining the empty set to be
openThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
. This empty topological space is the unique initial object in the
category of topological spacesIn 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 continuous...
with continuous maps.
Axiomatic set theory
In
Zermelo set theoryZermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
, the existence of the empty set is assured by the
axiom of empty setIn axiomatic set theory, the axiom of empty set is an axiom of Zermelo–Fraenkel set theory, the fragment thereof Burgess calls ST, and Kripke–Platek set theory.- Formal statement :...
, and its uniqueness follows from the
axiom of extensionalityIn 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.- Formal statement :...
. However, the axiom of empty set can be shown redundant in either of two ways:
- A logic such that provability and truth hold for both empty as well as nonempty domains is called a 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...
. Set theory is almost never formulated with free logic as its background logic; hence many theorems of set theory are valid only if the domain of discourseIn the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...
is nonempty. Canonical axiomatic set theory assumes that everything in the (nonempty) domain is a set. Therefore at least one set exists; call it A. By the axiom schema of separation (a theorem in some theories), the set B = {x | x∈A ∧ x ≠ x} exists and, having no members, is the empty set;
- The axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
, included in all mathematically interesting axiomatic set theories, not only asserts the existence of an infinite set I (from which B in the preceding paragraph may be constructed), but typically requires that the empty set be a member of I.
Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.
The empty set is not the same thing as
nothing; rather, it is a set with nothing
inside it and a set is always
something. 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 in
chessChess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
that involve a
kingIn chess, the king is the most important piece. The object of the game is to trap the opponent's king so that its escape is not possible . If a player's king is threatened with capture, it is said to be in check, and the player must remove the threat of capture on the next move. If this cannot be...
."
The popular
syllogismA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
- 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
" and the latter to "The set {ham sandwich} is better than the set
". It is noted that the first compares elements of sets, while the second compares the sets themselves.
Jonathan LoweJonathan Lowe is currently Professor of Philosophy and Chair of the Examination Board of the Department of Philosophy at Durham University, England. He was born in the UK, educated at the University of Cambridge, 1968-72, and the University of Oxford, 1972-75...
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 set which has no members. We cannot conjure such an entity into existence by mere stipulation."
George BoolosGeorge Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
argued that much of what has been heretofore obtained by set theory can just as easily be obtained by
plural quantificationIn 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 etc...
over individuals, without reifying sets as singular entities having other entities as members.
The empty set is a crucial part of the philosophy of
Alain BadiouAlain Badiou is a French philosopher, professor at European Graduate School, formerly chair of Philosophy at the École Normale Supérieure . Along with Giorgio Agamben and Slavoj Žižek, Badiou is a prominent figure in an anti-postmodern strand of continental philosophy...
. For Badiou, the empty set, referred to as "the void," is considered to be "the proper name of being" and thus the foundational material of
ontologyOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
.