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- Empty_function abstract "In mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty function The graph of an empty function is a subset of the Cartesian product ∅ × A. Since the product is empty the only such subset is the empty set ∅. The empty subset is a valid graph since for every x in the domain ∅ there is a unique y in the codomain A such that (x,y) ∈ ∅ × A. This statement is an example of a vacuous truth since there is no x in the domain.The existence of an empty function from ∅ to ∅ is required to make the category of sets a category, because in a category, each object needs to have an "identity morphism", and only the empty function is the identity on the object ∅.The existence of a unique empty function from ∅ into each set A means that the empty set is an initial object in the category of sets. In terms of cardinal arithmetic, it means that k0 = 1 for every cardinal number k – particularly profound when k = 0 to illustrate the strong statement of indices pertaining to 0.When defining the term "constant function" precisely, most authors will not care whether or not the empty function qualifies, and will use whatever definition is most convenient. Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations. This is much along the same lines of not considering 1 to be a prime number, an empty topological space to be connected, or the trivial group to be simple.".
- Empty_function wikiPageID "309274".
- Empty_function wikiPageRevisionID "540417351".
- Empty_function hasPhotoCollection Empty_function.
- Empty_function subject Category:Basic_concepts_in_set_theory.
- Empty_function subject Category:Functions_and_mappings.
- Empty_function subject Category:Nothing.
- Empty_function subject Category:Types_of_functions.
- Empty_function type Abstraction100002137.
- Empty_function type BasicConceptsInSetTheory.
- Empty_function type Cognition100023271.
- Empty_function type Concept105835747.
- Empty_function type Content105809192.
- Empty_function type Function113783816.
- Empty_function type FunctionsAndMappings.
- Empty_function type Idea105833840.
- Empty_function type MathematicalRelation113783581.
- Empty_function type PsychologicalFeature100023100.
- Empty_function type Relation100031921.
- Empty_function comment "In mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty function The graph of an empty function is a subset of the Cartesian product ∅ × A. Since the product is empty the only such subset is the empty set ∅. The empty subset is a valid graph since for every x in the domain ∅ there is a unique y in the codomain A such that (x,y) ∈ ∅ × A.".
- Empty_function label "Empty function".
- Empty_function label "Función vacía".
- Empty_function label "Funkcja pusta".
- Empty_function label "Funzione vuota".
- Empty_function label "Lege functie".
- Empty_function label "Пустая функция".
- Empty_function label "空関数".
- Empty_function sameAs Función_vacía.
- Empty_function sameAs Funzione_vuota.
- Empty_function sameAs 空関数.
- Empty_function sameAs Lege_functie.
- Empty_function sameAs Funkcja_pusta.
- Empty_function sameAs m.01sz53.
- Empty_function sameAs Q2295354.
- Empty_function sameAs Q2295354.
- Empty_function sameAs Empty_function.
- Empty_function wasDerivedFrom Empty_function?oldid=540417351.
- Empty_function isPrimaryTopicOf Empty_function.