Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set.
Null set was once a common synonym for "empty set", but is now a technical term in measure theory. The empty set may also be called the void set.
Notation
Common notations for the empty set include "{}", "∅", and "
". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets (and not related in any way to the Greek letter Φ).
The empty-set symbol ∅ is found at Unicode point U+2205. In TeX, it is coded as \emptyset or \varnothing.
Properties
In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements; therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set".
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