Image (mathematics)

Summary

is a function from domain to codomain The yellow oval inside is the image of

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

More generally, evaluating a given function 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 of is the set of all elements of the domain that map to the members of

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 "take the value " or "take as a value" if there exists some in the function's domain such that Similarly, given a set is said to "take a value in " if there exists some in the function's domain such that However, " takes [all] values in " and " is valued in " means that for every point in 's domain.

Image of a subset

The image of a subset under denoted is the subset of which can be defined using set-builder notation as follows:[1][2]

When there is no risk of confusion, is simply written as This convention is a common one; the intended meaning must be inferred from the context. This makes a function whose domain is the power set of (the set of all subsets of ), and whose codomain is the power set of See § Notation 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.[3] This 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 [4] 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

Notation for image and inverse image

The traditional notations used in the previous section may be confusing, because it does not distinguish the original function from the image-of-sets function ; likewise it does not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). An alternative[5] 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 [6][7]
  • Some texts refer to the image of as the range of [citation needed] but this usage should be avoided because the word "range" is also commonly used to mean the codomain of

Examples

  1. 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 is the empty set
  2. 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.
  3. defined by
    The fiber are concentric circles about the origin, the origin itself, and the empty set, depending on whether respectively. (if then the fiber is the set of all satisfying the equation of the origin-concentric ring )
  4. 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.
  5. A quotient group is a homomorphic image.

Properties

Counter-examples based on the real numbers
defined by
showing that equality generally need
not hold for some laws:
Image showing non-equal sets: The sets and are shown in blue immediately below the -axis while their intersection is shown in green.

General

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

Image Preimage

(equal if for instance, if is surjective)[8][9]

(equal if is injective)[8][9]
[8]
[10] [10]
[10] [10]

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
[10][11]
[10][11]
(equal if is injective[12])
[10]
(equal if is injective[12])
[10]

(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

  1. ^ "5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 2019-11-05. Retrieved 2020-08-28.
  2. ^ Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Sect.8
  3. ^ Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 2020-08-28.
  4. ^ Dolecki & Mynard 2016, pp. 4–5.
  5. ^ Blyth 2005, p. 5.
  6. ^ Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S.
  7. ^ M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
  8. ^ a b c See Halmos 1960, p. 39
  9. ^ a b See Munkres 2000, p. 19
  10. ^ a b c d e f g h See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
  11. ^ a b Kelley 1985, p. 85
  12. ^ a b See Munkres 2000, p. 21

References

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