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Codomain ○｜Definition｜1st｜20251119205401-00-⌔

Codomain - Wikipedia

Codomain

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In mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are 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 the 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.¹ 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.²³ 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.⁴

Printed 2026-06-28.

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Footnotes

1. Bourbaki 1970, p. 76 ↩

2. Bourbaki 1970, p. 77 ↩

3. Forster 2003, pp. 10–11 ↩

4. Eccles 1997, p. 91 (quote 1, quote 2); Mac Lane 1998, p. 8; Mac Lane, in Scott & Jech 1967, p. 232; Sharma 2004, p. 91; Stewart & Tall 1977, p. 89 ↩

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