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Note that a null set is not necessarily an empty set. 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 alphabets (and not related in any way to the Greek letter Φ). [2] Empty sets ...
In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set , while in other theories, its existence can be deduced.
In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. [3] [5] In measure theory, a null set is a (possibly nonempty) set with zero measure. A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element).
The standard construction of the Cantor set is an example of a null uncountable set in ; however other constructions are possible which assign the Cantor set any measure whatsoever. All the subsets of R n {\displaystyle \mathbb {R} ^{n}} whose dimension is smaller than n {\displaystyle n} have null Lebesgue measure in R n . {\displaystyle ...
The null morpheme is represented as either the figure zero (0) or the empty set symbol ∅. In most languages, it is the affixes that are realized as null morphemes, indicating that the derived form does not differ from the stem. For example, plural form sheep can be analyzed as combination of sheep with added null affix for the plural.
Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a ...
Compare this to the Scandinavian vowel "Ø", the "empty set" symbol "∅" and the diameter symbol ⌀. A convention common on early line printers left zero unornamented but added a tail or hook to the letter-O so that it resembled an inverted Q (like U+213A ℺ ) or cursive capital letter-O ( O {\displaystyle \,{\mathcal {O}}\,} ).
It is a useful concept in analysis, indicating lack of an element where one might be expected. It is usually written with the symbol "∅", in Unicode U+2205 ∅ EMPTY SET (∅, ∅, ∅, ∅). A common ad hoc solution is to use the Scandinavian capital letter Ø instead. There are several kinds of zero: