In mathematics, for a function
, 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 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
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
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
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
"Preimage" redirects here. For the cryptographic attack on hash functions, see
preimage attack.
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 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
[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
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
Counter-examples based on the real numbers  defined by  showing that equality generally need not hold for some laws:
|
|
|
|
General
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
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
|
|
|
[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
Notes
- ^ "5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 2019-11-05. Retrieved 2020-08-28.
- ^ Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Sect.8
- ^ Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 2020-08-28.
- ^ Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S.
- ^ M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
- ^ Hoffman, Kenneth (1971). Linear Algebra (2nd ed.). Prentice-Hall. p. 388.
- ^ a b c See Halmos 1960, p. 31
- ^ a b See Munkres 2000, p. 19
- ^ a b c d e f g h See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
- ^ a b Kelley 1985, p. 85
- ^ 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.