Empty set

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 (zero) members. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. In Set theory, the axiom of empty set is one of the Axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory Many possible properties of sets are trivially true for the empty set. In Mathematics, the term trivial is frequently used for objects (for examples groups or Topological spaces that have a very simple

Null set was once a common synonym for "empty set," but this usage should should be avoided because "null set" is now a technical term in measure theory. In Mathematics, a null set is a set that is negligible in some sense. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

1 Notation
2 Properties
- 2.1 Operations on the empty set
3 Mathematics
4 Does the empty set exist?
- 4.1 Axiomatic set theory
- 4.2 Philosophical issues
5 Use in linguistics
6 See also
7 Notes
8 References

Notation

A symbol for empty set

Common notations for the empty set include "{}," " $\varnothing,$ " and " $\emptyset.$ " The latter two symbols were introduced by the Bourbaki group (specifically Andre Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabet. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions The " Ø " ( minuscule: " ø " is a Vowel and a letter used in the Danish, Faroese and Norwegian The Danish and Norwegian Alphabet is based upon the Latin alphabet and has consisted of the following 29 letters since 1917 (Norwegian and 1955 ^[1] Other notations for the empty set include "Λ", "0", and "‣" ^[2]

Properties

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. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of extensionality, or axiom 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. This is a listing of common symbols found within all branches of the science of Mathematics.

For any set A:

The empty set is a subset of A:

∀A: ∅ ⊆ A
The union of A with the empty set is A:

∀A: A ∪ ∅ = A
The intersection of A with the empty set is the empty set:

∀A: A ∩ ∅ = ∅
The Cartesian product of A and the empty set is empty:

∀A: A × ∅ = ∅

The empty set has the following properties:

Its only subset is the empty set itself:

∀A: A ⊆ ∅ ⇒ A = ∅
The power set of the empty set is a set containing only the empty set:

2^∅ = {∅}
Its number of elements (that is, its cardinality) is zero. In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" Moreover, the empty set is finite:

|∅| = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, zero is defined as the empty set. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Several ways have been proposed to define the Natural numbers using Set theory.

For any property:

For every element of ∅ the property holds (vacuous truth);
There is no element of ∅ for which the property holds. In modern Philosophy, Mathematics, and Logic, a property is an Attribute of an object; thus a red object is said to have the property A 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

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 V = ∅.

By the definition of subset, 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 truth. A 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 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) can also be confusing. (Such operations are nullary operations. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function ) 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). Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” do not exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

Mathematics

Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then any member of that set will be an upper bound and lower bound for the empty set. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced Ordered set is used with distinct meanings in Order theory. A set with a Binary relation R on its elements that is reflexive (for 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. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a ^[3] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted $-\infty\!\,,$ which is defined to be less than every other extended real number, and positive infinity, denoted $+\infty\!\,,$ which is defined to be greater than every other extended real number, then:

$\sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,$

and

$\inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.$

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of 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 space), the empty set is both closed and open. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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. For a different notion of boundary related to Manifolds see that article In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Moreover, the empty set is a compact set by the fact that every finite set is compact. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2.

The closure of the empty set is empty. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set This is known as "preservation of nullary unions. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets "

Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an empty function is a function whose domain is the Empty set. As a result, the empty set is the unique initial object of the category of sets and functions. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

Does the empty set exist?

Axiomatic set theory

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. Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Set theory. In Set theory, the axiom of empty set is one of the Axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of extensionality, or axiom 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. Free logic is a Logic with no Existential Presuppositions Alternatively it is a logic whose theorems are valid in all domains including the Empty 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. The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive 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. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom schema of specification, axiom 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 In Set theory, an infinite set is a set that is not a Finite set.

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. In Philosophy, ontology (from the Greek, genitive: of being (part

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 can be a stumbling block. If so, the following homely figure of speech may be helpful. Think of a set as a bag, and its members as being the contents of the bag. An empty bag undoubtedly still exists.

Jonathan Lowe argues that while the empty set:

". Jonathan Lowe (EJ Lowe (born 1950 is currently Professor of Philosophy and Chair of the Examination Board of the Department of Philosophy at Durham University, England . . 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 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. George Stephen Boolos ( September 4, 1940, New York City – May 27, 1996) was a Philosopher and a Mathematical logician In Mathematics and logic, plural quantification is the theory that an individual Variable x may take on ^[4]

Tom McKay has disparaged the "singularist" assumption that natural expressions using plurals can be analysed using plural surrogates, such as signs for sets. He argues for an anti-singularist theory which differs from set theory in that there is no analogue of the empty set, and there is just one relation, among, that is an analogue of both the membership and the subset relation.

Use in linguistics

Set theory generally is a basic tool in formal semantics. See also Formal semantics of programming languages. Formal semantics is the study of the Semantics, or Interpretations Hence the empty set plays an important role in linguistics. It is used in language-teaching to denote a natural form (also colloquially named the dictionary form), which is generally the nominative singular for languages with declensions. The nominative case is a Grammatical case for a Noun, which generally marks the subject of a Verb, as opposed to its object or other In linguistics grammatical number is a Grammatical category of nouns pronouns and adjective and verb agreement that expresses count distinctions (such as "one" A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them In Linguistics, declension (or declination) is the occurrence of Inflection in Nouns Pronouns and Adjectives indicating It is also employed to emphasize that nothing should be added to the noun. However, this type of empty set is usually written with the same size as the other letters and so looks more like a ø than like a ∅.

The empty set symbol is sometimes used in natural language syntax and morphology to represent morphemes that are not pronounced. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the Morphology is the field of Linguistics that studies the internal structure of words

Notes

^ Earliest Uses of Symbols of Set Theory and Logic.
^ John B. Conway, Functions of One Complex Variable, 2nd ed. John Bligh Conway is a Mathematician at the George Washington University. P. 12.
^ Bruckner, A. N. , Bruckner, J. B. , and Thomson, B. S. , 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.
^ *George Boolos, 1984, "To be is to be the value of a variable," The Journal of Philosophy 91: 430-49. George Stephen Boolos ( September 4, 1940, New York City – May 27, 1996) was a Philosopher and a Mathematical logician Reprinted in his 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J. Richard C Jeffrey ( 5 August 1926 – 9 November 2002) was an American Philosopher, Logician, and probability , eds. ) Harvard Univ. Press: 54-72.

References

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.

Dictionary

-noun

(set theory) The unique set that contains no elements. Notation: Ø or {}

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