In mathematics, the empty set is the set with no elements.

Table of contents
1 Notation
2 Properties
3 Common problems
4 Axiomatic set theory
5 Does it exist or is it necessary?
6 Operations on the empty set
7 The empty set and zero
8 Category theory

Notation

Here we will notate the empty set {}, but another notable notation, invented by Bourbaki is the symbolism or ∅, which is approximated sometimes by the glyph "" and confused occasionally with the Greek letter "φ".

Properties

(Here we use mathematical symbolss.)

  • 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

We 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.

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.