Image (mathematics)

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

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

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.

DefinitionEdit

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

Image of an elementEdit

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 subsetEdit

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 functionEdit

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 relationsEdit

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 imageEdit

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 imageEdit

The traditional notations used in the previous section may be confusing[why?]. An alternative[5] is to give explicit names for the image and preimage as functions between power sets:

Arrow notationEdit

  •   with  
  •   with  

Star notationEdit

  •   instead of  
  •   instead of  

Other terminologyEdit

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

ExamplesEdit

  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.

PropertiesEdit

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.
 
 
 
 

GeneralEdit

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 functionsEdit

For functions   and   with subsets   and   the following properties hold:

  •  
  •  

Multiple subsets of domain or codomainEdit

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 alsoEdit

NotesEdit

  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

ReferencesEdit

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