codominio

Codominio, in mathematics, is the set that is designated as the target of a function. If a function f is defined as f: D -> C, then D is the domain (dominio) and C is the codomain (codominio). The function assigns to every element of the domain exactly one element of the codomain.

The image or range of a function is the subset of the codomain consisting of all actual outputs: Im(f) = { f(x) | x in D }. Thus, Im(f) is contained in the codomain, but it need not equal it. When the image equals the codomain, the function is surjective (onto); if the image is a proper subset, the function is not surjective.

Examples help clarify the distinction:

- Consider f: R -> R given by f(x) = x^2. The codomain is R, but the image is [0, ∞). This function is not onto R.

- Consider f: N -> N given by f(n) = n. Here the codomain and the image are both N, and f is bijective (in particular, surjective).

The codomain is part of the data that defines a function. Different choices of codomain for the same rule can change properties such as surjectivity, even though the rule itself remains unchanged. In category theory and related areas, the distinction between domain, codomain, and image becomes especially important for composing functions and discussing morphisms.