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In
mathematics, the '''empty set''' is the
set with no elements.
== Notation ==
The standard notation for denoting the empty set, invented by
Nicholas Bourbaki, is the symbol <math>\varnothing</math>, also written as <math>\emptyset</math> or ∅, and sometimes approximated by the glyph "
", (not to be confused with the
Greek letter "φ"). However, for wider browser compatibility this encyclopedia generally uses the notation "{}".
== Properties ==
(Here we use
mathematical symbols.)
*
For any set ''A'', the empty set is a
subset of ''A'':
*: ∀''A'': {} ⊆ ''A''
* For any set ''A'', the
union of ''A'' with the empty set is ''A'':
*: ∀''A'': ''A'' ∪ {} = ''A''
* For any set ''A'', the
intersection of ''A'' with the empty set is the empty set:
*: ∀''A'': ''A'' ∩ {} = {}
* For any set ''A'', the
cartesian product of ''A'' and the empty set is empty:
*: ∀''A'': ''A'' × {} = {}
* The only subset of the empty set is the empty set itself:
*: ∀''A'': ''A'' ⊆ {} ⇒ ''A'' = {}
* The
cardinality of the empty set is
zero; in particular, the empty set is
finite:
*: |{}| = 0
Mathematicians speak of "the empty set" rather than "an empty set". In set theory, two sets are equal if they have the same elements; therefore there can be only one set with no elements.
The empty set is both
closed and
open. The boundary points in it, which are empty, are in the empty set, and the set is therefore closed, while the interior points in it, which are empty again, are the subset of the empty set, and the set is therefore open. Moreover, the empty set is a
compact set by the fact every finite set is compact.
The
closure of the empty set is empty. This is known as "preservation of
nullary unions."
== Common problems ==
The empty set is not the same thing as "nothing"; it is a set with nothing ''in'' it, and a set is ''something''. This often causes difficulty among those who first encounter it. It may stem, in part, from the gap between intuitive structures that are generally modelled by sets, such as piles of objects, and the formal definition of a set. For example, we would not speak of a "pile of zero dishes", yet we will happily speak of a "set of zero elements", the empty set. It may then be helpful to think of a set as a bag containing its elements; an empty bag may be empty, but it certainly exists.
Some people balk at the first property listed above, that the empty set is a subset of any set ''A''. By the definition of
subset, this claim means that for ''every'' element ''x'' of {}, ''x'' belongs to ''A''. Since "every" is a strong word, we intuitively expect that it must be necessary to find ''many'' elements of {} that also belong to ''A'', but of course, we can't find ''any'' elements of {}, period. So you might think that {} is not a subset of ''A'' after all. But in fact, "every" is not a strong word at all when it appears in the phrase "every element of {}". 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, "every element of {}" does not actually refer to anything, and so there is no element of {} that is not in ''A'', leading us to conclude that every element of {} is in A and hence that {} 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".
== Axiomatic set theory ==
In the
axiomatization of set theory known as
Zermelo-Fraenkel set theory, the existence of the empty set is assured by the
axiom of empty set. The uniqueness of the empty set follows from the
axiom of extensionality.
== Does it exist or is it necessary? ==
While the empty set is a standard and universally accepted concept in mathematics, there are those who still entertain doubts.
Jonathan Lowe has argued that while the idea "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 not clear that such an idea makes sense. "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".
In "To be is to be the value of a variable …",
Journal of Philosophy , 1984 (reprinted in his book ''Logic, Logic and Logic''), the late
George Boolos has argued that we can go a long way just by
quantifying plurally over individuals, without
reifying sets as singular entities having other entities as members.
In a recent book
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.
== 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''.)
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” don't 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.
== The empty set and zero ==
It was mentioned earlier that the empty set has
zero elements, or that its cardinality is zero. The connection between the two concepts goes further however: in the standard
set-theoretic definition of natural numbers, zero is ''defined'' as the empty set.
== Category theory ==
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 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.
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