Editing Empty set (section)

== Properties ==
In standard [[axiomatic set theory]], by the [[axiom of extensionality|principle of extensionality]], two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

The only subset of the empty set is the empty set itself; equivalently, the [[power set]] of the empty set is the set containing only the empty set.  The number of elements of the empty set (i.e., its [[cardinality]]) is zero.  The empty set is the only set with either of these properties.

[[For any]] set ''A'':
* The empty set is a [[subset]] of ''A''
* The [[union (set theory)|union]] of ''A'' with the empty set is ''A''
* The [[intersection (set theory)|intersection]] of ''A'' with the empty set is the empty set
* The [[Cartesian product]] of ''A'' and the empty set is the empty set

For any [[property (philosophy)|property]] ''P'':
* For every element of <math>\varnothing</math>, the property ''P'' holds ([[vacuous truth]]).
* There is no element of <math>\varnothing</math> for which the property ''P'' holds.

Conversely, if for some property ''P'' and some set ''V'', the following two statements hold:
* For every element of ''V'' the property ''P'' holds
* There is no element of ''V'' for which the property ''P'' holds
then <math>V = \varnothing.</math>

By the definition of [[subset]], the empty set is a subset of any set ''A''. That is, {{em|every}} element ''x'' of <math>\varnothing</math> belongs to ''A''. Indeed, if it were not true that every element of <math>\varnothing</math> is in ''A'', then there would be at least one element of <math>\varnothing</math> that is not present in ''A''. Since there are {{em|no}} elements of <math>\varnothing</math> at all, there is no element of <math>\varnothing</math> that is not in ''A''. Any statement that begins "for every element of <math>\varnothing</math>" 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."

In the usual [[set-theoretic definition of natural numbers]], zero is modelled by the empty set.

=== Operations on the empty set ===

When speaking of the [[summation|sum]] of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the [[empty sum]]) is zero. The reason for this is that zero is the [[identity element]] for addition. Similarly, the [[multiplication|product]] of the elements of the empty set (the [[empty product]]) should be considered to be [[1 (number)|one]], since one is the identity element for multiplication.<ref>{{cite book |author=David M. Bloom |title=Linear Algebra and Geometry |url=https://archive.org/details/linearalgebrageo0000bloo |url-access=registration |year=1979 |isbn=0521293243 |pages=[https://archive.org/details/linearalgebrageo0000bloo/page/45 45]}}</ref>

A [[derangement]] is a [[permutation]] of a set without [[fixed point (mathematics)|fixed point]]s. The empty set can be considered a derangement of itself, because it has only one permutation (<math>0!=1</math>), and it is vacuously true that no element (of the empty set) can be found that retains its original position.