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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, the image of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is the set of all output values it may produce. More generally, evaluating a given function f at each
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
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 *Doma ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of 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 Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
, not just functions.


Definition

The word "image" is used in three related ways. In these definitions, f : X \to Y is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from the set X to the set Y.


Image of an element

If 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: * Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them ** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...
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 f's domain.


Image of a subset

Throughout, let f : X \to Y be a function. The under f of a subset A of 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 In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
, this definition can be written as f = \. This induces a function f ,\cdot\,: \wp(X) \to \wp(Y), where \wp(S) denotes the
power set Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...
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 u ...

subset
s of S. See below for more.


Image of a function

The ''image'' of a function is the image of its entire
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 *Doma ...
, 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 ...
of the function. This usage should be avoided because the word "range" is also commonly used to mean the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of f.


Generalization to binary relations

If 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 elem ...
on 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 to 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, denoted by f^ /math> or by f^ is also called the
fiber Fiber or fibre (from la, fibra, links=no) is a #Natural fibers, natural or #Man-made fibers, man-made substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineer ...
or fiber over y or the
level set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
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, an inverse function (or anti-function) is a function (mathematics), function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i ...
, 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 may be confusing, because it does not distinguish the original function f : X \to Y from the image-of-sets function f : \mathcal(X) \to \mathcal(Y); 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

* 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 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 formal syst ...
and
set theory illustrating the intersection of two sets 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 ...
is f\,''A.M. Randall Holmes
Inhomogeneity of the urelements in the usual models of NFU
December 29, 2005, on: Semantic Scholar, p. 2
* 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of f.


Examples

# f : \ \to \ defined by f(x) = \left\{\begin{matrix} a, & \mbox{if }x=1 \\ a, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \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}(\{ 1, 2 \}) = \{ 1, 2 \}. The preimage of \{ b, d \}, is the
empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical an ...

empty set
\{ \, \} = \varnothing. # f : \R \to \R defined by f(x) = x^2. The ''image'' of \{ -2, 3 \} under f is f^{-1}(\{ -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 ''fiber'' f^{-1}(\{ a \}) are
concentric circles In geometry, two or more mathematical object, objects are said to be concentric, coaxal, or coaxial when they share the same center (geometry), center or Coordinate axis, axis. Circles, regular polygons and regular polyhedron, regular polyhedra, ...

concentric circles
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 Sla ...
, the origin itself, and the
empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical an ...

empty set
, depending on whether a > 0, a = 0, \text{ or } a < 0, respectively. (if a > 0, then the fiber f^{-1}(\{ a \}) is the set of all (x, y) \in \R^2 satisfying the equation of the origin-concentric ring x^2 + y^2 = a.) # If M is a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
and \pi : TM \to M is the canonical projection from the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

tangent bundle
TM to M, then the ''fibers'' of \pi are the
tangent spaces In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
T_x(M) \text{ for } x \in M. This is also an example of a
fiber bundle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
. # A
quotient group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...
is a homomorphic image.


Properties

{, class=wikitable style="float:right;" , + ! Counter-examples based on the
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s \R,
f : \R \to \R defined by x \mapsto x^2,
showing that equality generally need
not hold for some laws: , - , , - , center, upright=1.7, f\left(f^{-1}\left(B_3\right)\right) \subsetneq B_3. , - ,


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 BSee 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 The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.) With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a
semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
homomorphism (that is, it does not always preserve intersections).


See also

* * * *


Notes


References

* * . * * * * {{PlanetMath attribution, id=3276, title=Fibre
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...
Isomorphism theorems