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In mathematics, the empty set is the set with no elements.
Notation
Here we will denote the empty set by {}, but another notable notation, invented by Bourbaki, is the symbol ,
also written as or ∅, and sometimes
approximated by the glyph "Ø", and occasionally confused with the Greek letter "φ".
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 {}". Since there are no elements of {}, "every element of {}" does not actually refer to anything, so
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 is (1) 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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