In mathematics, and more specifically set theory, the empty set is the unique set having no (zero) members. 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 this usage should be avoided because "null set" is now a technical term in measure theory.
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Common notations for the empty set include "{}," "" and "
" 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.[1] Other notations for the empty set include "Λ", "0", and "‣" [2]
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 has the following properties:
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:
Conversely, if for some property and some set V, the following two statements hold:
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. This is often paraphrased as "everything is true of the elements of 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.) 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). 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.
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. 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.[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 which is defined to be less than every other extended real number, and positive infinity, denoted
which is defined to be greater than every other extended real number, then:
and
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.
Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open. 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."
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.
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:
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 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:
it is also the case that:
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.[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.
Set theory generally is a basic tool in formal semantics. 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. 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.