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In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size is zero. Some axiomatic set theories assure 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.
Contents
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 Danish and Norwegian alphabet (and not related in any way to the Greek letter Φ).^{[1]} Other notations for the empty set include "Λ", "0"^{[2]}
The emptyset symbol ∅ is found at Unicode point U+2205.^{[3]} In TeX, it is coded as \emptyset or \varnothing.
Properties
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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