Codomænet

Codomænet is a term used in some mathematical texts to refer to what is commonly called the codomain of a function. In a function f from a set X to a set Y, the codomænet is the set Y—the target set that contains all possible outputs the function could produce, regardless of which outputs actually occur.

The distinction between the codomænet and the image (often called the range in informal use) is fundamental. The image of f is the subset of the codomænet that consists of all values f(x) as x runs over X. The image may be smaller than the codomænet; if every element of the codomænet is attained by some input, the function is surjective (onto). If not, it may fail to be surjective even though the mapping is well-defined.

Examples illustrate the idea. Consider f: R → R defined by f(x) = x^2. The codomænet is R, but the image is [0, ∞). Hence f is not surjective onto R, though it is a valid function. Another example: g: Z → Z with g(n) = n + 1 has codomænet Z and image Z, so g is surjective.

Notes on use and terminology. Some texts use codomain and codomænet interchangeably, while others prefer to reserve codomænet for the target set in the formal definition of a function. Changing the codomænet can alter whether a function is surjective, even though the mapping itself remains the same.

See also: Domain, Image, Function, Surjection.