Image (mathematics)
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 "" 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 's domain.

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

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 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  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 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 )
 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).

See also

Notes

References

 
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Basic concepts in set theory
Isomorphism theorems