codomein

Codomein (codomain) is a term in mathematics describing the target set of a function. Given a function f from X to Y, denoted f: X -> Y, the codomein is the set Y that contains all potential outputs of f. The actual outputs realized by f on elements of X form its image (also called the range), which is a subset of the codomein. The distinction matters: a function can have a codomein larger than its image, in which case f is not surjective onto the codomein.

Examples help illustrate the concept. The function f: R -> R defined by f(x) = x^2 has codomein R but image [0, ∞). It is not surjective onto R; if instead the codomein is taken to be [0, ∞), then f is surjective. Another example is a function f: {1,2,3} -> {a,b} with 1 -> a, 2 -> a, 3 -> b; the codomein is {a,b} and the image is {a,b}, so f is surjective onto its codomein.

Interpretation and use: The codomein is part of the data that defines a function and influences properties such as surjectivity. Changing the codomein can change whether a function is surjective. In function composition, the codomein of the first function must be compatible with the domain of the next function. In programming languages, the concept corresponds to the return type, which constrains what values a function may produce.