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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.
Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object. Morphisms of pointed sets.
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 ...
The empty set is a subset of every set (the statement that all elements of the empty set are also members of any set A is vacuously true). The set of all subsets of a given set A is called the power set of A and is denoted by or (); the "P" is sometimes in a script font: ℘ .
A function that, given a set of non-empty sets, assigns to each set an element from that set. Fundamental in the formulation of the axiom of choice in set theory. choice negation In logic, an operation that negates the principles underlying the axiom of choice, exploring alternative set theories where the axiom does not hold. choice set
For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.
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In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
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