A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "input" arguments in a function. For any given function , the set A, on which f is defined (the arguments), is called the domain, and B (the set of possible values) is called the codomain of f. The set of all actual values or f(A) is called the range of f. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Partial plot of a function f. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. The codomain is not to be confused with the range, which is in general only a subset of B; in lower-level mathematics education, however, range is often taught as being equivalent to codomain. In mathematics, the range of a function is the set of all output values produced by that function. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... A lecture on linear algebra Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. ...

Example

Let the function f be a function on the real numbers: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

defined by

The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R₀⁺—non-negative reals, i.e. the interval [0,∞): A negative number is a number that is less than zero, such as −3. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

One could have defined the function g thus:

While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.

The codomain can affect whether the function is a surjection; in our example, g is a surjection while f is not. The codomain does not affect whether the function is an injection. A surjective function. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

See also

• Domain (mathematics)
• Range (mathematics)