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# Empty set

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

 Table of contents 1 Notation 2 Properties 3 Philosophical Difficulties 4 Operations on the empty set 5 Category theory

## Notation

The empty set is written either as "Ø" (which derives from the Norwegian letter "Ø" and is sometimes conflated with the Greek letter "φ") or simply as "{}" (which is the preferred symbol in this encyclopedia).

## Properties

• For any set A, the empty set is a subset of A:
{} ⊆ A
• For any set A, the union of A with the empty set is A:
A ∪ {} = A
• For any set A, the intersection of A with the empty set is the empty set:
A ∩ {} = {}
• The only subset of the empty set is the empty set itself.
• The cardinality of the empty set is zero; in particular, the empty set is finite:
|{}| = 0

## Philosophical Difficulties

While the empty set is a standard and universally accepted concept in mathematics, there are those who still entertain doubts. 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. And as Jonathan Lowe has pointed out, it would be at best a bad joke "to be told that, say, there is a number of pound notes in a sealed envelope that has just been given to them, when in fact the envelope is empty. The response "Well, I did say a number of pound notes, and nought is a number" would do nothing to pacify the irate recipient."

Lowe argues 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 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.

## 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 initial object of the category of sets and functions.

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