> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/260-function-theory/topics/codomain.md). # Codomain The codomain of a function , or the target set, is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of 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 has elements y in its codomain for which the equation f(x) = y does not have a solution. A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (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 f: X → Y. *** A codomain of a function is the target set in operations or relations on two sets. Given two sets `A` and `B`, where `∀a ∈ A` and `∀b ∈ B`, set `B` is the codomain: * in set operations (union, intersection, etc.), e.g. `A U B = {a ∨ b}` * in the dot product, `A x B = {(a,b)}` * in relations on sets, `aRb` i.e. `{(a,b)} = R ⊆ A x B` * in functions, `f :: A -> B` and `f = {(a,b)}` The codomain set is sometimes denoted by $$B\_{cod}$$.