Empty set

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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 "{}," "\varnothing", and "\emptyset". 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 empty-set 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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