Image (mathematics)

, the image of an input value

is the single output value produced by

The preimage of an output value

is the set of input values that produce

produces a set, called the "image of

Similarly, the inverse image (or preimage) of a given subset

is the set of all output values it may produce, that is, the image of

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 the function's domain such that

Using set-builder notation, this definition can be written as[1][2]

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

[3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of

is an arbitrary binary relation on

is called the image, or the range, of

The preimage or inverse image of a set

[4] The inverse image of a singleton set, denoted by

The set of all the fibers over the elements of

is a family of sets indexed by

can also be thought of as a function from the power set of

should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of

The traditional notations used in the previous section do not distinguish the original function

; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets).

Given the right context, this keeps the notation light and usually does not cause confusion.

But if needed, an alternative[5] is to give explicit names for the image and preimage as functions between power sets: For every function

the following properties hold: Also: For functions

the following properties hold: For function

the following properties hold: 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: (Here,

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 (that is, 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.

For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.
is a function from domain to codomain . The image of element is element . The preimage of element is the set { }. The preimage of element is .
is a function from domain to codomain . The image of all elements in subset is subset . The preimage of is subset
is a function from domain to codomain The yellow oval inside is the image of . The preimage of is the entire domain
Image showing non-equal sets: The sets and are shown in blue immediately below the -axis while their intersection is shown in green .