In mathematics, the image of a function is the set of all such that belongs to the domain of . The image by of an element of the domain of is , that is, the output corresponding to the input . The image by of a subset of the domain of is the set of all such that is in , that is, the set of the images of the elements of . Equivalently, it is the image of the restriction of to .

Preimages or inverse images are defined similarly, by exchanging the roles of the domain and the codomain:

The preimage of an element of the codomain of is the set of all elements of the domain of such that ; it is empty if does not belong to the image of . The preimage of a subset of the codomain of is the set of all elements of the domain of such that . The preimage of the codomain of is, by definition of a function, the domain of .

Images and inverse images may also be defined similarly for general binary relations. in this generalization, images and preimages play symmetric roles: the images and the preimages of a relation are respectively the preimages and the images of the opposite relation.

Definition

is a function from the set.

Image of an element

, is the value.

is said to or is said to if there exists . However, and for .

Image of a subset

be a function. The when there is no risk of confusion. Using set-builder notation, this definition can be written as

denotes the power set; that is the set of all subsets. See below for more.

Image of a function

The image of a function is the image of its entire domain, also known as the range of the function. This last usage should be avoided because the word "range" is also commonly used to mean the codomain.

Generalization to binary relations

is an arbitrary binary relation is called is called.

Inverse image

The preimage or inverse image

The inverse image of a singleton set is also called the fiber, or the level set is a family of sets

should not be confused with that for inverse function

Notation for image and inverse image

; 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 is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

Star notation

Other terminology

• used in mathematical logic and set theory

• but this usage should be avoided because the word "range" is also commonly used to mean the codomain

Examples

1. The image The image The preimage The preimage The preimage is the empty set

2. The image and the image (the set of all positive real numbers and zero). The preimage The preimage is the empty set, because the negative numbers do not have square roots in the set of reals.

3. The fibers are concentric circles about the origin, the origin itself, and the empty set then the fiber)

4. is a manifold is the canonical projection from the tangent bundle then the fibers are the tangent spaces This is also an example of a fiber bundle.

5. A quotient group is a homomorphic image.

Properties

General

the following properties hold:Also:

Multiple functions

the following properties hold:

Multiple subsets of domain or codomain

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:

can be infinite, even uncountably infinite.)

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).

See also

Notes

References

• .