Any set other than the empty set is callednon-empty.
In some textbooks and popularizations, the empty set is referred to as the "null set".[1] However,null set is a distinct notion within the context ofmeasure theory, in which it describes a set of measure zero (which is not necessarily empty).
Common notations for the empty set include "{ }", "", and "∅". The latter two symbols were introduced by theBourbaki group (specificallyAndré Weil) in 1939, inspired by the letterØ (U+00D8ØLATIN CAPITAL LETTER O WITH STROKE) in theDanish andNorwegian alphabets.[2] In the past, "0" (the numeralzero) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.[3]
The symbol ∅ is available atUnicode pointU+2205∅EMPTY SET.[4] It can be coded inHTML as∅ and as∅ or as∅. It can be coded inLaTeX as\varnothing. The symbol is coded in LaTeX as\emptyset.
When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode characterU+29B0⦰REVERSED EMPTY SET may be used instead.[5]
In standardaxiomatic set theory, by theprinciple 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, thepower set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., itscardinality) is zero. The empty set is the only set with either of these properties.
For every element of, the propertyP holds (vacuous truth).
There is no element of for which the propertyP holds.
Conversely, if for some propertyP and some setV, the following two statements hold:
For every element ofV the propertyP holds
There is no element ofV for which the propertyP holds
then
By the definition ofsubset, the empty set is a subset of any setA. That is,every elementx of belongs toA. Indeed, if it were not true that every element of is inA, then there would be at least one element of that is not present inA. Since there areno elements of at all, there is no element of that is not inA. Any statement that begins "for every element of" is not making any substantive claim; it is avacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."
When speaking of thesum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (theempty sum) is zero. The reason for this is that zero is theidentity element for addition. Similarly, theproduct of the elements of the empty set (theempty product) should be considered to beone, since one is the identity element for multiplication.[6]
Aderangement is apermutation of a set withoutfixed points. The empty set can be considered a derangement of itself, because it has only one permutation (), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
Since the empty set has no member when it is considered as a subset of anyordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by thereal number line, every real number is both an upper and lower bound for the empty set.[7] When considered as a subset of theextended reals formed by adding two "numbers" or "points" to the real numbers (namelynegative infinity, denoted which is defined to be less than every other extended real number, andpositive infinity, denoted which is defined to be greater than every other extended real number), we have that:and
That is, the least upper bound (sup orsupremum) of the empty set is negative infinity, while the greatest lower bound (inf orinfimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.
In anytopological space, the empty set isopen by definition, as is. Since thecomplement of an open set isclosed and the empty set and are complements of each other, the empty set is also closed, making it aclopen set. Moreover, the empty set iscompact by the fact that everyfinite set is compact.
A topological space is said to have theindiscrete topology if the only open sets are and the entire space.
In thevon Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as. Thus, we have,,, and so on. The von Neumann construction, along with theaxiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers,, such that thePeano axioms of arithmetic are satisfied.
In the context of sets of real numbers, Cantor used to denote " contains no single point". This notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed as an existent set on its own, or if Cantor merely used as an emptiness predicate. Zermelo accepted itself as a set, but considered it an "improper set".[9]
Standardfirst-order logic implies, merely from thelogical axioms, thatsomething exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from theaxiom of separation.
Even usingfree logic (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely theaxiom of infinity.
While the empty set is a standard and widely accepted mathematical concept, it remains anontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.
The empty set is not the same thing asnothing; rather, it is a set with nothinginside it and a set is alwayssomething. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of allopening moves inchess that involve aking."[10]
Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is" and the latter to "The set {ham sandwich} is better than the set". The first compares elements of sets, while the second compares the sets themselves.[10]
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 also the case that:
"All that we are ever informed about the empty set is that it (1) is 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, aset which has no members. We cannot conjure such an entity into existence by mere stipulation."[11]
George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained byplural quantification over individuals, withoutreifying sets as singular entities having other entities as members.[12]
