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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 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 * ...
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 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, 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

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: 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 ...
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, this definition can be written as f = \. This induces a function f ,\cdot\,: \mathcal P(X) \to \mathcal P(Y), where \mathcal P(S) denotes the power set of a set S; that is the set of all
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
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 * ...
, 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 of f.


Generalization to binary relations

If R is an arbitrary binary relation 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, 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 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, 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 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 forma ...
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 concern ...
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 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 \{ \ \} = \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 Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
, the origin itself, and the empty set (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 and \pi : TM \to M is the canonical projection from the tangent bundle TM to 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. # A quotient group 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 ...
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 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 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.) 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 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