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In , and more specifically , the empty set is the unique set having no elements; its size or (count of elements in a set) is zero. Some axiomatic set theories ensure 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 for the empty set.

was once a common synonym for "empty set", but is now a technical term in and describes a set that is not necessarily empty. The empty set may also be called the void set.


Notation
Common notations for the empty set include "{}", "\emptyset", and "∅". The latter two symbols were introduced by the (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets (and not related in any way to the Greek letter Φ). Earliest Uses of Symbols of Set Theory and Logic. 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.
(1976). 007054235X, McGraw-Hill. . 007054235X

The symbol ∅ is available at point U+2205. Unicode Standard 5.2 It can be coded in as ∅ and in as \varnothing. The symbol \emptyset is coded in LaTeX as \emptyset; it is not available in HTML/Unicode.


Properties
In standard axiomatic set theory, 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.

set A:

  • The empty set is a of A:
  • :\forall A: \varnothing \subseteq A
  • The union of A with the empty set is A:
  • :\forall A: A \cup \varnothing = A
  • The intersection of A with the empty set is the empty set:
  • :\forall A: A \cap \varnothing = \varnothing
  • The Cartesian product of A and the empty set is the empty set:
  • :\forall A: A \times \varnothing = \varnothing

The empty set has the following properties:

  • Its only subset is the empty set itself:
  • :\forall A: A \subseteq \varnothing \Rightarrow A = \varnothing
  • The of the empty set is the set containing only the empty set:
  • :2^{\varnothing } = \{\varnothing\}
  • Its number of elements (that is, its ) is zero:
  • :\mathrm{card}(\varnothing) = 0

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

For any property:

  • For every element of \varnothing the property holds ();
  • There is no element of \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 V = \varnothing.

By the definition of , the empty set is a subset of any set A. That is, every element x of \varnothing belongs to A. Indeed, if it were not true that every element of \varnothing is in A then there would be at least one element of \varnothing that is not present in A. Since there are no elements of \varnothing at all, there is no element of \varnothing that is not in A. Any statement that begins "for every element of \varnothing" is not making any substantive claim; it is a . This is often paraphrased as "everything is true of the elements of the empty set."


Operations on the empty set
When speaking of the 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 for addition. Similarly, the of the elements of the empty set should be considered to be one (see ), since one is the identity element for multiplication.

A is a of a set without fixed 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

Extended real numbers
Since the empty set has no members, when it is considered as a subset of any , 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.Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9. When considered as a subset of the 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 ) of the empty set is negative infinity, while the greatest lower bound (inf or ) 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
In any topological space X, the empty set is by definition, as is X. Since the complement of an open set is and the empty set and X are complements of each other, the empty set is also closed, making it a . Moreover, the empty set is by the fact that every is compact.

The closure of the empty set is empty. This is known as "preservation of unions."


Category theory
If A is a set, then there exists precisely one function f from {} to A, the . As a result, the empty set is the unique of the 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 . This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: 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 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 the axiom 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 the of arithmetic are satisfied.


Questioned existence

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. However, the axiom of empty set can be shown redundant in either of two ways:
  • There is already an axiom implying the existence of at least one set. Given such an axiom together with the axiom of separation, the existence of the empty set is easily proved.
  • In the presence of , it is easy to prove that at least one set exists, viz. the set of all urelements (assuming there is not a proper class of them). Again, given the axiom of separation, the empty set is easily proved.


Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an 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 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 in that involve a king."

The popular

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". It is noted that the first compares elements of sets, while the second compares the sets themselves.
(2018). 9780471270478, John Wiley and Sons.

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."

argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without sets as singular entities having other entities as members.*, 1984, "To be is to be the value of a variable," The Journal of Philosophy 91: 430–49. Reprinted in his 1998 Logic, Logic and Logic (, and Burgess, J., eds.) Harvard Univ. Press: 54–72.


See also


Further reading
  • , Naive Set Theory. 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).


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