Empty set
In
Any set other than the empty set is called non-empty.
In some textbooks and popularizations, the empty set is referred to as the "null set".
Notation
Common notations for the empty set include "{ }", "", and "∅". The latter two symbols were introduced by the
The symbol ∅ is available at Unicode point U+2205 ∅ EMPTY SET.[4] It can be coded in HTML as ∅
and as ∅
or as ∅
. It can be coded in LaTeX as \varnothing
. The symbol is coded in LaTeX as \emptyset
.
When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.[5]
Properties
In standard
The only subset of the empty set is the empty set itself; equivalently, the power set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its cardinality) is zero. The empty set is the only set with either of these properties.
- The empty set is a subset of A
- The union of A with the empty set is A
- The intersection of A with the empty set is the empty set
- The Cartesian product of A and the empty set is the empty set
For any property P:
- For every element of , the property P holds (vacuous truth).
- There is no element of for which the property P holds.
Conversely, if for some property P and some set V, the following two statements hold:
- For every element of V the property P holds
- There is no element of V for which the property P holds
then
By the definition of subset, the empty set is a subset of any set A. That is, every element x of belongs to A. Indeed, if it were not true that every element of is in A, then there would 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. 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."
In the usual set-theoretic definition of natural numbers, zero is modelled by the empty set.
Operations on the empty set
When speaking of the
A derangement is a permutation of a set without fixed points. The empty set can be considered a derangement of itself, because it has only one permutation (), 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 member when it is considered as a subset of any
That is, the least upper bound (sup or
Topology
In any
The
Category theory
If is a set, then there exists precisely one function from to the
The empty set can be turned into a
Set theory
In the
Questioned existence
Historical issues
In the context of sets of real numbers, Cantor used to denote " contains no single point". This notation was utilized in definitions, for example Cantor defined two sets as being disjoint if their intersection has an absence of points, however it is debatable whether Cantor viewed as an existent set on its own, or if Cantor merely used as an emptiness predicate. Zermelo accepted itself as a set, but considered it an "improper set".[7]
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 at least two ways:
- Standard axiom of separation.
- Even using free logic (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the axiom of infinity.
Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an
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."[8]
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 " and the latter to "The set {ham sandwich} is better than the set ". The first compares elements of sets, while the second compares the sets themselves.[8]
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."[9]
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.[10]
See also
- 0 – Number
- Inhabited set – Property of sets used in constructive mathematics
- Nothing – Complete absence of anything; the opposite of everything
- Power set – Mathematical set containing all subsets of a given set
References
- ^ a b Weisstein, Eric W. "Empty Set". mathworld.wolfram.com. Retrieved 2020-08-11.
- ^ "Earliest Uses of Symbols of Set Theory and Logic".
- ISBN 007054235X.
- ^ "Unicode Standard 5.2" (PDF).
- ^ e.g. Nina Grønnum (2005, 2013) Fonetik og Fonologi: Almen og dansk. Akademisk forlag, Copenhagen.
- ^ Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). Elementary Real Analysis, 2nd edition, p. 9.
- ^ A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
- ^ ISBN 0-471-27047-4.
- ^ E. J. Lowe (2005). Locke. Routledge. p. 87.
- ^ George Boolos (1984), "To be is to be the value of a variable", The Journal of Philosophy 91: 430–49. Reprinted in 1998, Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard University Press, 54–72.
Further reading
- ISBN 978-1-61427-131-4(paperback edition).
- ISBN 3-540-44085-2
- Graham, Malcolm (1975), Modern Elementary Mathematics (2nd ed.), ISBN 0155610392