Image (mathematics)

In mathematics, for a function , the image is a relation between inputs and outputs, used in three related ways:

The image of an input value is the single output value produced by when passed . The preimage of an output value is the set of input values that produce .
More generally, evaluating at each element of a given subset of its domain produces a set, called the "image of under (or through) ". Similarly, the inverse image (or preimage) of a given subset of the codomain is the set of all elements of that map to a member of
The image of the function is the set of all output values it may produce, that is, the image of . The preimage of is the preimage of the codomain . Because it always equals (the domain of ), it is rarely used.

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

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

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 fiber 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

<span id="Notation">Notation</span> 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
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

General

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

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:

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).

Notes

References

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