Editing Codomain (section)

{{Short description|Target set of a mathematical function}}
[[File:Codomain2.SVG|right|thumb|250px|A function {{mvar|f}} from {{mvar|X}} to {{mvar|Y}}. The blue oval {{mvar|Y}} is the codomain of {{mvar|f}}. The yellow oval inside {{mvar|Y}} is the [[Image (mathematics)|image]] of {{mvar|f}}, and the red oval {{mvar|X}} is the [[Domain of a function|domain]] of {{mvar|f}}.]]

In [[mathematics]], a '''codomain''', '''counter-domain''', or '''set of destination''' of a [[Function (mathematics)|function]] is a [[Set (mathematics)|set]] into which all of the output of the function is constrained to fall. It is the set {{mvar|Y}} in the notation {{math|''f'': ''X'' → ''Y''}}. The term '''''[[Range of a function|range]]''''' is sometimes ambiguously used to refer to either the codomain or the [[Image (mathematics)|''image'']] of a function.

A codomain is part of a function  {{mvar|f}}  if  {{mvar|f}}  is defined as a triple {{math|(''X'', ''Y'', ''G'')}} where {{mvar|X}} is called the ''[[Domain of a function|domain]]'' of {{mvar|f}}, {{mvar|Y}} its ''codomain'', and {{mvar|G}} its ''[[Graph of a function|graph]]''.<ref>{{Harvnb|Bourbaki|1970|p=76}}</ref> The set of all elements of the form {{math|''f''(''x'')}}, where {{mvar|x}} ranges over the elements of the domain {{mvar|X}}, is called the ''[[Image (mathematics)|image]]'' of {{mvar|f}}. The image of a function is a [[subset]] of its codomain so it might not coincide with it. Namely, a function that is not [[Surjective function|surjective]] has elements {{mvar|y}} in its codomain for which the equation {{math|1=''f''(''x'') = ''y''}} does not have a solution.

A codomain is not part of a function  {{mvar|f}}  if  {{mvar|f}}  is defined as just a graph.<ref>{{Harvnb|Bourbaki|1970|p=77}}</ref><ref>{{Harvnb|Forster|2003}}, [{{Google books|plainurl=y|id=mVeTuaRwWssC|page=10|text=Some mathematical cultures make this explicit, saying that a function}} pp. 10&ndash;11]</ref> For example in [[set theory]] it is desirable to permit the domain of a function to be a [[Class (set theory)|proper class]] {{mvar|X}}, in which case there is formally no such thing as a triple {{math|(''X'', ''Y'', ''G'')}}. With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form  {{math|''f'': ''X'' → ''Y''}}.<ref>{{Harvnb|Eccles|1997}}, p. 91 ([{{Google books|plainurl=y|id=ImCSX_gm40oC|page=91|text=The reader may wonder at this variety of ways of thinking about a function}} quote 1], [{{Google books|plainurl=y|id=ImCSX_gm40oC|page=91|text=When defining a function using a formula it is important to be clear about which sets are the domain and the codomain of the function}} quote 2]); {{Harvnb|Mac Lane|1998}}, [{{Google books|plainurl=y|id=MXboNPdTv7QC|page=8|text=Here "function" means a function with specified domain and specified codomain}} p. 8]; Mac Lane, in {{Harvnb|Scott|Jech|1967}}, [{{Google books|plainurl=y|id=5mf4Vckj0gEC|page=232|text=Note explicitly that the notion of function is not that customary in axiomatic set theory}} p. 232]; {{Harvnb|Sharma|2004}}, [{{Google books|plainurl=y|id=IGvDpe6hYiQC|page=91|text=Functions as sets of ordered pairs}} p. 91]; {{Harvnb|Stewart|Tall|1977}}, [{{Google books|plainurl=y|id=TLelvnIU2sEC|page=89|text=Strictly speaking we cannot talk of 'the' codomain of a function}} p. 89]</ref>