The empty set is the set containing no elements.

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set.

Null set was once a common synonym for "empty set", but is now a technical term in measure theory.

Notation[link]

A symbol for the empty set

Common notations for the empty set include "{}," "Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing ", and "Failed to parse (Missing texvc executable; please see math/README to configure.): \emptyset ". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabet (and not related in any way to the Greek letter Φ).^[1] Other notations for the empty set include "Λ" and "0"^[2]

The empty-set symbol ∅ is found at Unicode point U+2205.^[3] In TeX, it is coded as \emptyset or \varnothing.

Properties[link]

By the principle of extensionality, 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 here.

For any set A:

• The empty set is a subset of A:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \forall A: \varnothing \subseteq A\, .

• The union of A with the empty set is A:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \forall A: A \cup \varnothing = A\, .

• The intersection of A with the empty set is the empty set:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \forall A: A \cap \varnothing = \varnothing\, .

• The Cartesian product of A and the empty set is empty:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \forall A: A \times \varnothing = \varnothing\, .

The empty set has the following properties:

• Its only subset is the empty set itself:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \forall A: A \subseteq \varnothing \Rightarrow A = \varnothing\, .

• The power set of the empty set is a set containing only the empty set:

  Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{\varnothing} = \{\varnothing\}\, .

• Its number of elements (that is, its cardinality) is zero:

  Failed to parse (Missing texvc executable; please see math/README to configure.): | \varnothing | = 0\, .

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.

For any property:

• For every element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing

the property holds (vacuous truth);

• There is no element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing

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 Failed to parse (Missing texvc executable; please see math/README to configure.): V = \varnothing

.

By the definition of subset, the empty set is a subset of any set A, as every element x of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing

belongs to A. If it is not true that every element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
is in A, there must be at least one element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
that is not present in A. Since there are no elements of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
at all, there is no element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
that is not in A. Hence every element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
is in A, and Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing
is a subset of A. Any statement that begins "for every element of Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing

" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."

Operations on the empty set[link]

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the identity element for addition, and one is the identity element for multiplication.

Mathematics[link]

Extended real numbers[link]

Since the empty set has no members, when it is considered as a subset of any ordered set, 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.^[4] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): -\infty\!\,,

which is defined to be less than every other extended real number, and positive infinity, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): +\infty\!\,,
which is defined to be greater than every other extended real number, then: 

Failed to parse (Missing texvc executable; please see math/README to configure.): \sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) 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[link]

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open; it is an example of a "clopen" set. All its boundary points (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 closure of the empty set is empty. This is known as "preservation of nullary unions."

Category theory[link]

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps.

Questioned existence[link]

Axiomatic set theory[link]

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. 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. 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 discourse 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, 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[link]

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 chess that involve a king."^[5]

The popular syllogism

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing " and the latter to "The set {ham sandwich} is better than the set Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing ". It is noted that the first compares elements of sets, while the second compares the sets themselves.^[5]

Jonathan Lowe 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."^{[citation needed]}

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.^[6]

The empty set is a crucial part of the philosophy of Alain Badiou. 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 ontology.

See also[link]

• Inhabited set

Notes[link]

References[link]

• Paul Halmos, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.

Look up empty in Wiktionary, the free dictionary.

Empty may refer to:

• Empty (God Live Underwater album) (1995)
• Empty (Tait album) (2001)
• "Empty" (The Click Five song)
• "Empty" (Metric song)
• "Empty", a song by the Cranberries from No Need to Argue
• "Empty", a song by Ray LaMontagne from Till The Sun Turns Black
• Empty (TV series), a 2008 situation comedy appearing on BBC Two
• Empty (magazine), an Australian Magazine

See also[link]

• Emptiness (disambiguation)

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Look up set in Wiktionary, the free dictionary.

Set may refer to:

Mathematics and programming[link]

• Set (mathematics), A collection of well defined and distinct objects
• Set theory
• Category of sets
• Set (computer science), a data type that is a collection of unique values
  • Set (C++), an implementation of Set (computer science) in the C++ language Standard Template Library (STL)

Chemistry[link]

• SET, single electron transfer

Psychology[link]

• Set (psychology), a set of expectations which shapes perception or thought

Technology[link]

• Set, to make/become solid: see Solidification
• A television set
• SET, Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks
• SET, Single-Electron Transistor, a device to amplify currents in nanoelectronics
• SET, single-ended triode, a type of electronic amplifier
• SET Awards, The Science, Engineering & Technology Student of the Year Awards
• Set (command), a command for manipulating environment variables in Unix and Microsoft operating-systems
• Software Engineer in Test, a Quality Assurance job title in some software companies
• Saw set, the process of setting the teeth of a saw so each tooth protrudes to the side of the blade

Arts[link]

Theatre[link]

• Theatrical scenery
• Set construction, construction of scenery for theatrical, movie and video game production.

Dancing[link]

• Set, the basic square formation in square dancing

Music[link]

• DJ set or DJ mix, a musical performance by a DJ
• Set theory (music), dealing with concepts for categorizing musical objects and describing their relationships
• Set (music), a collection of discrete entities, for example pitch sets, duration sets, and timbre sets
• Set (Thompson Twins album)
• Set (Alex Chilton album)
• Set list, a list of songs to be performed at a concert
• In the professional jargon of music production and critique, a "set" refers to an album of songs.

Fiction[link]

• Set or Father Set, a fictional deity in the Conan adventure novels by Robert E. Howard and their Marvel Comics interpretation

Sport and games[link]

• A chess set
• Set, a signal used in American football
• Set (darts), when one player wins three legs
• Set (game), a card game
• Set, a unit of play in tennis
• Set, a term for three of a kind in poker
• Set, a unit of play in volleyball (formerly "game")
• Set, a group of repetitions in weight training
• The set, a pitching position in baseball (also called the stretch)

Crime[link]

• Set, a sub-group of a street gang

Religion[link]

• Set or Seth, a Biblical character, a son of Adam and Eve
• Set (mythology), an ancient Egyptian deity

Miscellaneous[link]

• Set and setting, coined by Timothy Leary to describe the mindset and location of hallucinogenic experiences
• Seṭ and aniṭ roots, in Sanskrit grammar
• Set (video game), a group of items that adds specific bonuses
• Set (river), a river in northeastern Spain
• Set or sett, a badger's den
• Set, a small tuber or bulb used instead of seed, especially:
  • Potato set
  • Onion set

Other acronyms[link]

• Sanlih Entertainment Television, a television channel in Taiwan
• Scalar expectancy theory
• Serial endosymbiosis theory, a theory concerning the origins of mitochondria and plastids
• Similarity Enhanced Transfer, a technology used to speed up download rate of files on file sharing networks
• Simulated Emergency Test, an amateur radio training exercise
• Single Embryo Transfer
• Societatea Pentru Exploatări Tehnice, a Romanian aeronautics company of the 1920s to 1940s
• Society of Engineering Technologists, a student organization at Texas Tech University
• Sony Entertainment Television, a television channel owned by Sony
• South Eastern Trains, a rail company operating in South East England
• Stock Exchange of Thailand, the national stock exchange of Thailand
  • SET Index, an index for the Stock Exchange of Thailand
• Strategic Energy Technologies Plan (EU SET Plan)
• Study of Exceptional Talent, a program for gifted students
• Single-Ended Triode, a type of electronic amplifier
• Social entropy Theory, application of entropy concept to maintenance or disorganisation in society
• Northeastern University, School of Engineering Technology
• Southeast Toyota Distributors LLC, a distributor of Toyota vehicles
• Suzuki SET, Suzuki Exhaust Tuning, a device for altering pressure waves within motorcycle exhausts

See also[link]

• Sett (disambiguation)
• Setting (disambiguation)
• Setup (disambiguation)
• Seti (disambiguation)

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.