mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the image of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

is the set of all output values it may produce. More generally, evaluating a given function

f

at each element of a given subset

A

of its

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

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, the codomain or set of destination of a function is the 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 refer to either the ...

f,

is the set of all elements of the domain that map to the members of

B.

Image and inverse image may also be defined for general

binary relations In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

, not just functions.

Definition

The word "image" is used in three related ways. In these definitions,

f : X \to Y

is a

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 Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...

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

f

's domain.

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, 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 are ...

s of

S.

See below for more.

Image of a function

The ''image'' of a function is the image of its entire

, also known as the

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

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 elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

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 ...

, denoted by

f^/math> or by f^is also called the

fiber Fiber or fibre (from la, fibra, links=no) 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 incorporate ...

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 calle ...

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 X\t ...

, 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 within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

and

set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...

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 is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

\{ \ \} = \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 In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center poi ...

about the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

, 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, projections or projective 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, graphic ...

from the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

TM

M,

then the ''fibers'' of

\pi

are the

tangent spaces In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to ...

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 (or, in 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 p ...

. # 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 examp ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

\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 and

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

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 (or uncountably infinite set) 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 num ...

.) With respect to the algebra of subsets described above, the inverse image function is a

lattice homomorphism A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...

, 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 a ...

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