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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 are used in set operations.
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 logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
A set is empty if the sentence () is true, where the notation is shorthand for (). If L {\displaystyle L} is any set then the following are equivalent: L {\displaystyle L} is not empty, meaning that the sentence ¬ [ ∀ x ( x ∉ L ) ] {\displaystyle \lnot [\forall x(x\not \in L)]} is true (literally, the logical negation of " L {\displaystyle ...
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).
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( ) sets and it is by definition equal to the empty set.
Now consider two subsets of S and set their distance apart as the size of their symmetric difference. This distance is in fact a metric, which makes the power set on S a metric space. If S has n elements, then the distance from the empty set to S is n, and this is the maximum distance for any pair of subsets. [6]
braces, a.k.a. curly brackets (for set notation) 1895 Georg Cantor: Blackboard bold capital N (for natural numbers set) 1895 Giuseppe Peano ... empty set sign 1939