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Empty set

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

Table of contents

Notation

The empty set is written either as "" (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

Common misconceptions

It should be noted that the empty set is not the same thing as "nothing"; it is a set with nothing in it, and a set is something. In fact, precise application of the empty set can help one understand the various notions of "nothing" as that word is used in natural language. For example, consider this classic joke:

  • Nothing is better than eternal happiness.
  • But a ham sandwich is better than nothing.
  • Therefore, a ham sandwich is better than eternal happiness.
We all recognise the logic in this joke as nonsensical, but it may not be obvious how the two uses of "nothing" can be reconciled. The first statement says:
The set of things that are better than eternal happiness is {}.
The second statement says:
The set {ham sandwich} is better than the set {}.
Now we can see that the two sentences aren't related; the first is comparing individual things, while the second is comparing sets of things, and {} (the set of "nothing") plays completely different roles in them.

Ironically, while the empty set can be used to analyse the intuitive concept of "nothing", the concept of the empty set itself often causes a degree of confusion among those who first encounter it. This 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 probably tend not to speak of a "pile of zero pencils", yet we will happily speak of a "set of zero elements", the empty set.

For example, 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. See Vacuous truth for more about this logical phenomenon.

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 emtpy set is the initial object of the category of sets and functions.



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