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

f: X \to Y

, the image of an input value

x

is the single output value produced by

f

when passed

x

. The preimage of an output value

y

is the set of input values that produce

y

. More generally, evaluating

f

at each element of a given subset

A

of its domain

X

produces a set, called the "image of

A

under (or through)

f

". Similarly, the inverse image (or preimage) of a given subset

B

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 ...

Y

is the set of all elements of

X

that map to a member of

B.

The image of the function

f

is the set of all output values it may produce, that is, the image of

X

. The preimage of

f

is the preimage of the codomain

Y

. Because it always equals

X

(the domain of

f

), 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,

f : X \to Y

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 ...

X

to the set

Y.

Image of an element

x

is a member of

X,

then the image of

x

under

f,

denoted

f(x),

is the value of

f

when applied to

x.

f(x)

is alternatively known as the output of

f

for argument

x.

Given

y,

the function

f

is said to or if there exists some

x

in the function's domain such that

f(x) = y.

Similarly, given a set

S,

f

is said to if there exists

x

in the function's domain such that

f(x) \in S.

However, and means that

f(x) \in S

for point

x

in the domain of

f

Image of a subset

Throughout, let

f : X \to Y

be a function. The under

f

of a subset

A

X

is the set of all

f(a)

for

a\in A.

It is denoted by

f

or by

f(A)

when there is no risk of confusion. Using set-builder notation, this definition can be written as

f = \.

This induces a function

: \mathcal P(X) \to \mathcal P(Y),

where

\mathcal P(S)

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

S;

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

S.

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

f.

Generalization to binary relations

R

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 ...

X \times Y,

then the set

\

is called the image, or the range, of

R.

Dually, the set

\

is called the domain of

R.

Inverse image

Let

f

be a function from

X

Y.

The preimage or inverse image of a set

B \subseteq Y

under

f,

denoted by

f^

is the subset of

X

defined by

f^B = \.

Other notations include

f^(B)

and

f^(B).

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

f^/math> or by f^(y), is also called the

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 ...

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 ...

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

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

f(\{ 2, 3 \}) = \{ a, c \}.

The ''image'' of the function

f

\{ a, c \}.

The ''preimage'' of

a

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

f(\{ -2, 3 \}) = \{ 4, 9 \},

and the ''image'' of

f

\R^+

(the set of all positive real numbers and zero). The ''preimage'' of

\{ 4, 9 \}

under

f

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

(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

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 ...

\R,

f : \R \to \R

defined by

x \mapsto x^2,

showing that equality generally need
not hold for some laws: , - , Image preimage conterexample intersection

, - , , - , Image preimage conterexample fb

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 injectiveSee ) ,

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).

Notes

References

* * . * * * * {{PlanetMath attribution, id=3276, title=Fibre Basic concepts in set theory Isomorphism theorems

Definition

Image of an element

Image of a subset

Image of a function

Generalization to binary relations

Inverse image

Notation for image and inverse image

Arrow notation

Star notation

Other terminology

Examples

Properties

General

Multiple functions

Multiple subsets of domain or codomain

See also

Notes

References