
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
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
In mathematics, a binary relation associates some elements of one set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is ...
, not just functions.
Definition

The word "image" is used in three related ways. In these definitions,
is a
function from the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 the domain of
.
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
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of a set
that is the set of all
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s 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
In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
of
Generalization to binary relations
If
is an arbitrary
binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
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
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the a ...
, denoted by
fiber
Fiber (spelled fibre in British English; from ) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often inco ...
or fiber over
y or the
level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~.
When the number of independent variables is two, a level set is call ...
of
y. The set of all the fibers over the elements of
Y is a family of sets indexed by
Y.
For example, for the function
f(x) = x^2, the inverse image of
\ would be
\. Again, if there is no risk of confusion,
f^ /math> can be denoted by f^(B), and f^ can also be thought of as a function from the power set of Y to the power set of X. The notation f^ should not be confused with that for inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
, although it coincides with the usual one for bijections in that the inverse image of
B under
f is the image of
B under
f^.
Notation for image and inverse image
The traditional notations used in the previous section do not distinguish the original function
f : X \to Y from the image-of-sets function
f : \mathcal(X) \to \mathcal(Y); 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
*
f^\rightarrow : \mathcal(X) \to \mathcal(Y) with
f^\rightarrow(A) = \
*
f^\leftarrow : \mathcal(Y) \to \mathcal(X) with
f^\leftarrow(B) = \
Star notation
*
f_\star : \mathcal(X) \to \mathcal(Y) instead of
f^\rightarrow
*
f^\star : \mathcal(Y) \to \mathcal(X) instead of
f^\leftarrow
Other terminology
* An alternative notation for
f /math> used in mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
and
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
is
f\,''A.
* Some texts refer to the image of
f as the range of
f, but this usage should be avoided because the word "range" is also commonly used to mean the
codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
of
f.
Examples
#
f : \ \to \ defined by
\left\{\begin{matrix}
1 \mapsto a, \\
2 \mapsto a, \\
3 \mapsto c.
\end{matrix}\right.
The ''image'' of the set
\{ 2, 3 \} under
f is
f(\{ 2, 3 \}) = \{ a, c \}. The ''image'' of the function
f is
\{ a, c \}. The ''preimage'' of
a is
f^{-1}(\{ a \}) = \{ 1, 2 \}. The ''preimage'' of
\{ a, b \} is also
f^{-1}(\{ a, b \}) = \{ 1, 2 \}. The ''preimage'' of
\{ b, d \} under
f is the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
\{ \ \} = \emptyset.
#
f : \R \to \R defined by
f(x) = x^2. The ''image'' of
\{ -2, 3 \} under
f is
f(\{ -2, 3 \}) = \{ 4, 9 \}, and the ''image'' of
f is
\R^+ (the set of all positive real numbers and zero). The ''preimage'' of
\{ 4, 9 \} under
f is
f^{-1}(\{ 4, 9 \}) = \{ -3, -2, 2, 3 \}. The ''preimage'' of set
N = \{ n \in \R : n < 0 \} under
f is the empty set, because the negative numbers do not have square roots in the set of reals.
#
f : \R^2 \to \R defined by
f(x, y) = x^2 + y^2. The
''fibers'' f^{-1}(\{ a \}) are
concentric circles about the
origin, the origin itself, and the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
(respectively), depending on whether
a > 0, \ a = 0, \text{ or } \ a < 0 (respectively). (If
a \ge 0, then the ''fiber''
f^{-1}(\{ a \}) is the set of all
(x, y) \in \R^2 satisfying the equation
x^2 + y^2 = a, that is, the origin-centered circle with radius
\sqrt{a}.)
# If
M is a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
and
\pi : TM \to M is the canonical
projection
Projection or projections may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphics, and carto ...
from the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
TM to
M, then the ''fibers'' of
\pi are the
tangent spaces T_x(M) \text{ for } x \in M. This is also an example of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
.
# A
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
is a homomorphic ''image''.
Properties
{, class=wikitable style="float:right;"
, +
! Counter-examples based on the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
\R, f : \R \to \R defined by
x \mapsto x^2, showing that equality generally need
not hold for some laws:
, -
,

, -
,
, -
,
General
For every function
f : X \to Y and all subsets
A \subseteq X and
B \subseteq Y, the following properties hold:
{, class="wikitable"
, -
! Image
! Preimage
, -
,
f(X) \subseteq Y
,
f^{-1}(Y) = X
, -
,
f\left(f^{-1}(Y)\right) = f(X)
,
f^{-1}(f(X)) = X
, -
,
f\left(f^{-1}(B)\right) \subseteq B(equal if
B \subseteq f(X); for instance, if
f is surjective)
[See ][See ]
,
f^{-1}(f(A)) \supseteq A(equal if
f is injective)
[
, -
, f(f^{-1}(B)) = B \cap f(X)
, \left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)
, -
, f\left(f^{-1}(f(A))\right) = f(A)
, f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)
, -
, f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing
, f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)
, -
, f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B
, f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B
, -
, f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)
, f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X
, -
, f(X \setminus A) \supseteq f(X) \setminus f(A)
, f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)][
, -
, f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B][See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.]
, f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)[
, -
, f\left(A \cap f^{-1}(B)\right) = f(A) \cap B][
, f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)][
Also:
* f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing
]
Multiple functions
For functions f : X \to Y and g : Y \to Z with subsets A \subseteq X and C \subseteq Z, the following properties hold:
* (g \circ f)(A) = g(f(A))
* (g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))
Multiple subsets of domain or codomain
For function f : X \to Y and subsets A, B \subseteq X and S, T \subseteq Y, the following properties hold:
{, class="wikitable"
, -
! Image
! Preimage
, -
, A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)
, S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)
, -
, f(A \cup B) = f(A) \cup f(B)
, f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)
, -
, f(A \cap B) \subseteq f(A) \cap f(B)
(equal if f is injective[See ])
, f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)
, -
, f(A \setminus B) \supseteq f(A) \setminus f(B)
(equal if f is injective[)
, f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)][
, -
, f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)]
(equal if f is injective)
, f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)
, -
The results relating images and preimages to the ( Boolean) algebra of intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
and union work for any collection of subsets, not just for pairs of subsets:
* f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)
* f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f\left(A_s\right)
* f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)
* f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)
(Here, S can be infinite, even uncountably infinite
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
.)
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
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has ...
homomorphism (that is, it does not always preserve intersections).
See also
*
*
*
*
*
Notes
References
*
* .
*
*
*
*
{{PlanetMath attribution, id=3276, title=Fibre
Basic concepts in set theory
Isomorphism theorems