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

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

Image of an element

If is a member of , then the image of under , denoted , is the value of when applied to . is alternatively known as the output of for argument .

Given , the function is said to or if there exists some in the function's domain such that . Similarly, given a set is said to if there exists in the function's domain such that . However, and means that for point in the domain of .

Image of a subset

Throughout, let be a function. The under of a subset of is the set of all for . It is denoted by , or by when there is no risk of confusion. Using set-builder notation, this definition can be written as Here: Sect.8

This induces a function , where denotes the power set of a set ; that is the set of all subsets of . 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 of .

Generalization to binary relations

If is an arbitrary binary relation on , then the set is called the image, or the range, of . Dually, the set is called the domain of .

Inverse image

Let be a function from to The preimage or inverse image of a set under denoted by is the subset of defined by

Other notations include and The inverse image of a singleton set, denoted by or by is also called the fiber (or fibre) over , or the level set of The set of all the fibers over the elements of is a family of sets indexed by

For example, for the function the inverse image of would be Again, if there is no risk of confusion, can be denoted by and can also be thought of as a function from the power set of to the power set of The notation should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of under is the image of under

Notation for image and inverse image The traditional notations used in the previous section do not distinguish the original function from the image-of-sets 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 is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

with with

Star notation

instead of instead of

Other terminology

An alternative notation for used in mathematical logic and set theory is M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2 Some texts refer to the image of as the range of but this usage should be avoided because the word "range" is also commonly used to mean the codomain of

Examples defined by The image of the set under is The image of the function is The preimage of is The preimage of is also The preimage of under is the empty set defined by The image of under is and the image of is (the set of all positive real numbers and zero). The preimage of under is The preimage of set under is the empty set, because the negative numbers do not have square roots in the set of reals. defined by The fibers are concentric circles about the origin, the origin itself, and the empty set (respectively), depending on whether (respectively). (If then the fiber is the set of all satisfying the equation that is, the origin-centered circle with radius ) If is a manifold and is the canonical projection from the tangent bundle to then the fibers of are the tangent spaces This is also an example of a fiber bundle. A quotient group is a homomorphic image.

Properties

+ Counter-examples based on the real numbers defined by showing that equality generally neednot hold for some laws:

General

For every function and all subsets and the following properties hold:

Image Preimage(equal if for instance, if is surjective)See See (equal if is injective)See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.

Also:

Multiple functions

For functions and with subsets and the following properties hold:

Multiple subsets of domain or codomain

For function and subsets and the following properties hold:

Image Preimage(equal if is injectiveSee )(equal if is injective)(equal if is injective)

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, 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

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