codomainn.

1. Originally: the set of all elements related to a set of referents by a binary relation (see relation n. 2b). In later use: the second of the two sets forming the Cartesian product of which a given binary relation is defined as a subset. Now somewhat rare.For example, the codomain of the relation ‘is the mother of’ is the set of all children.A binary relation between the sets X and Y is formally defined as any set of ordered pairs ( x, y ) where x is an element of a set X and y is an element of a set Y. Under this definition, the codomain is the subset of Y consisting of all elements y which appear in the set of ordered pairs defining the relation.

2. A set specified in the definition of a function as being that from which the independent variable of the function takes its values.A function does not necessarily map elements of its domain (domain n. 4e) to each element of its codomain; for instance, the function defined by f: ℝ ⟶ ℝ, f( x) = x², where ℝ is the set of real numbers, only takes values from the set of non-negative real numbers.Although occurring earlier in this sense than in that relating to category theory (sense 3), the initially much more common use of codomain in category theory may have influenced wider adoption of the term in relation to functions.

3. Any object in a category that is the second object in an ordered pair defining a morphism, regarded as an abstraction of the codomain of a function.Before 1965, such an object was typically called the range of the morphism, mimicking the then current terminology from function theory. See, for instance, quot. 1963.