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.

Definition edit

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 edit

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 the domain of   .

Image of a subset edit

Throughout, let   be a function. The image 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[1][2]

 

This induces a function   where   denotes the power set of a set   that is the set of all subsets of   See § Notation below for more.

Image of a function edit

The image of a function is the image of its entire domain, also known as the range of the function.[3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of  

Generalization to binary relations edit

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 edit

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 edit

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

Arrow notation edit

  •   with  
  •   with  

Star notation edit

  •   instead of  
  •   instead of  

Other terminology edit

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

Examples edit

  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   under   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 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  )
  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 edit

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 edit

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

Image Preimage
   
   
 
(equal if   for instance, if   is surjective)[9][10]
 
(equal if   is injective)[9][10]
   
   
   
   
   
   [9]
 [11]  [11]
 [11]  [11]

Also:

  •  

Multiple functions edit

For functions   and   with subsets   and   the following properties hold:

  •  
  •  

Multiple subsets of domain or codomain edit

For function   and subsets   and   the following properties hold:

Image Preimage
   
 [11][12]  
 [11][12]
(equal if   is injective[13])
 
 [11]
(equal if   is injective[13])
 [11]
 
(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 edit

  • Bijection, injection and surjection – Properties of mathematical functions
  • Fiber (mathematics) – Set of all points in a function's domain that all map to some single given point
  • Image (category theory) – term in category theory
  • Kernel of a function – Equivalence relation expressing that two elements have the same image under a function
  • Set inversion – Mathematical problem of finding the set mapped by a specified function to a certain range

Notes edit

  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. ^ Hoffman, Kenneth (1971). Linear Algebra (2nd ed.). Prentice-Hall. p. 388.
  9. ^ a b c See Halmos 1960, p. 31
  10. ^ a b See Munkres 2000, p. 19
  11. ^ a b c d e f g h See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
  12. ^ a b Kelley 1985, p. 85
  13. ^ a b See Munkres 2000, p. 21

References edit

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