# Codomain

## Target set of a mathematical function / From Wikipedia, the free encyclopedia

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In mathematics, a **codomain** or **set of destination** of a function is a 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 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.^{[2]}^{[3]} 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]}