Image (mathematics)

In mathematics, the image of a function is the set of all output values it may produce.

More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f ". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B.

Image and inverse image may also be defined for general binary relations, not just functions.

The word "image" is used in three related ways. In these definitions, f : XY is a function from the set X to the set Y.

If x is a member of X, then the image of x under f, denoted f(x),[1] is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x.

The image of a function is the image of its entire domain, also known as the range of the function.[4]

The traditional notations used in the previous section can be confusing. An alternative[7] is to give explicit names for the image and preimage as functions between power sets:

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:

With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (i.e., it does not always preserve intersections).

This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.