Image (mathematics)

For other uses, seeImage (disambiguation).

$f$is a function from domain$X$to codomain$Y$.The image of element$x$is element$y$.The preimage of element$y$is the set {$x,x'$}. The preimage of element$y'$is$\varnothing$.

$f$is a function from domain$X$to codomain$Y$.The image of all elements in subset$A$is subset$B$.The preimage of$B$is subset$C$

$f$is a function from domain$X$to codomain$Y.$The yellow oval inside$Y$is the image of$f$.The preimage of$Y$is the entire domain$X$

Inmathematics,for a function$f:X\to Y$,theimageof an input value$x$is the single output value produced by$f$when passed$x$.Thepreimageof an output value$y$is the set of input values that produce$y$.

More generally, evaluating$f$at eachelementof a given subset$A$of itsdomain$X$produces a set, called the "imageof$A$under (or through)$f$".Similarly, theinverse image(orpreimage) of a given subset$B$of thecodomain$Y$is the set of all elements of$X$that map to a member of$B.$

Theimageof the function$f$is the set of all output values it may produce, that is, the image of$X$.Thepreimageof$f$,that is, the preimage of$Y$under$f$,always equals$X$(thedomainof$f$); therefore, the former notion is rarely used.

Image and inverse image may also be defined for generalbinary relations,not just functions.

Definition

The word "image" is used in three related ways. In these definitions,$f:X\to Y$is afunctionfrom theset$X$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 thevalueof$f$when applied to$x.$$f(x)$is alternatively known as the output of$f$for argument$x.$

Given$y,$the function$f$is said totake the value$y$ortake$y$as a valueif there exists some$x$in the function's domain such that$f(x)=y.$ Similarly, given a set$S,$$f$is said totake a value in$S$if there existssome$x$in the function's domain such that$f(x)\in S.$ However,$f$takes [all] values in$S$and$f$is valued in$S$means that$f(x)\in S$foreverypoint$x$in the domain of$f$.

Image of a subset

Throughout, let$f:X\to Y$be a function. Theimageunder$f$of a subset$A$of$X$is the set of all$f(a)$for$a\in A.$It is denoted by$f[A],$or by$f(A),$when there is no risk of confusion. Usingset-builder notation,this definition can be written as $$f[A]=\{f(a):a\in A\}.$$

This induces a function$f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),$where${\mathcal {P}}(S)$denotes thepower setof a set$S;$that is the set of allsubsetsof$S.$See§ Notationbelow for more.

Image of a function

Theimageof a function is the image of its entiredomain,also known as therangeof the function.This last usage should be avoided because the word "range" is also commonly used to mean thecodomainof$f.$

Generalization to binary relations

If$R$is an arbitrarybinary relationon$X\times Y,$then the set$\{y\in Y:xRy{\text{ for some }}x\in X\}$is called the image, or the range, of$R.$Dually, the set$\{x\in X:xRy{\text{ for some }}y\in Y\}$is called the domain of$R.$

Inverse image

"Preimage" redirects here. For the cryptographic attack on hash functions, seepreimage attack.

Let$f$be a function from$X$to$Y.$Thepreimageorinverse imageof a set$B\subseteq Y$under$f,$denoted by$f^{-1}[B],$is the subset of$X$defined by $$f^{-1}[B]=\{x\in X\,:\,f(x)\in B\}.$$

Other notations include$f^{-1}(B)$and$f^{-}(B).$ The inverse image of asingleton set,denoted by$f^{-1}[\{y\}]$or by$f^{-1}[y],$is also called thefiberor fiber over$y$or thelevel setof$y.$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$\{4\}$would be$\{-2,2\}.$Again, if there is no risk of confusion,$f^{-1}[B]$can be denoted by$f^{-1}(B),$and$f^{-1}$can also be thought of as a function from the power set of$Y$to the power set of$X.$The notation$f^{-1}$should not be confused with that forinverse 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$f^{-1}.$

Notationfor 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$f:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)$;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 alternativeis to give explicit names for the image and preimage as functions between power sets:

Arrow notation

• $f^{\rightarrow }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)$with$f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}$
• $f^{\leftarrow }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)$with$f^{\leftarrow }(B)=\{a\in X\;|\;f(a)\in B\}$

Star notation

• $f_{\star }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)$instead of$f^{\rightarrow }$
• $f^{\star }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)$instead of$f^{\leftarrow }$

Other terminology

• An alternative notation for$f[A]$used inmathematical logicandset theoryis$f\,''A.$
• 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 thecodomainof$f.$

Examples

For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela and Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.

1. $f:\{1,2,3\}\to \{a,b,c,d\}$defined by$\left\{{\begin{matrix}1\mapsto a,\\2\mapsto a,\\3\mapsto c.\end{matrix}}\right.$
   Theimageof the set$\{2,3\}$under$f$is$f(\{2,3\})=\{a,c\}.$Theimageof the function$f$is$\{a,c\}.$Thepreimageof$a$is$f^{-1}(\{a\})=\{1,2\}.$Thepreimageof$\{a,b\}$is also$f^{-1}(\{a,b\})=\{1,2\}.$Thepreimageof$\{b,d\}$under$f$is theempty set$\{\ \}=\emptyset.$
2. $f:\mathbb {R} \to \mathbb {R}$defined by$f(x)=x^{2}.$
   Theimageof$\{-2,3\}$under$f$is$f(\{-2,3\})=\{4,9\},$and theimageof$f$is$\mathbb {R} ^{+}$(the set of all positive real numbers and zero). Thepreimageof$\{4,9\}$under$f$is$f^{-1}(\{4,9\})=\{-3,-2,2,3\}.$Thepreimageof set$N=\{n\in \mathbb {R}:n<0\}$under$f$is the empty set, because the negative numbers do not have square roots in the set of reals.
3. $f:\mathbb {R} ^{2}\to \mathbb {R}$defined by$f(x,y)=x^{2}+y^{2}.$
   Thefibers$f^{-1}(\{a\})$areconcentric circlesabout theorigin,the origin itself, and theempty set(respectively), depending on whether$a>0,\ a=0,{\text{ or }}\ a<0$(respectively). (If$a\geq 0,$then thefiber$f^{-1}(\{a\})$is the set of all$(x,y)\in \mathbb {R} ^{2}$satisfying the equation$x^{2}+y^{2}=a,$that is, the origin-centered circle with radius${\sqrt {a}}.$)
4. If$M$is amanifoldand$\pi:TM\to M$is the canonicalprojectionfrom thetangent bundle$TM$to$M,$then thefibersof$\pi$are thetangent spaces$T_{x}(M){\text{ for }}x\in M.$This is also an example of afiber bundle.
5. Aquotient groupis a homomorphicimage.

Properties

See also:List of set identities and relations § Functions and sets

Counter-examples based on thereal numbers$\mathbb {R},$ $f:\mathbb {R} \to \mathbb {R}$defined by$x\mapsto x^{2},$ showing that equality generally need not hold for some laws:
Image showing non-equal sets:$f\left(A\cap B\right)\subsetneq f(A)\cap f(B).$The sets$A=[-4,2]$and$B=[-2,4]$are shown inblueimmediately below the$x$-axis while their intersection$A_{3}=[-2,2]$is shown ingreen.
$f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.$
$f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.$

General

For every function$f:X\to Y$and all subsets$A\subseteq X$and$B\subseteq Y,$the following properties hold:

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)	$f^{-1}(f(A))\supseteq A$ (equal if$f$is injective)
$f(f^{-1}(B))=B\cap f(X)$	$\left(f\vert _{A}\right)^{-1}(B)=A\cap f^{-1}(B)$
$f\left(f^{-1}(f(A))\right)=f(A)$	$f^{-1}\left(f\left(f^{-1}(B)\right)\right)=f^{-1}(B)$
$f(A)=\varnothing \,{\text{ if and only if }}\,A=\varnothing$	$f^{-1}(B)=\varnothing \,{\text{ if and only if }}\,B\subseteq Y\setminus f(X)$
$f(A)\supseteq B\,{\text{ if and only if }}{\text{ there exists }}C\subseteq A{\text{ such that }}f(C)=B$	$f^{-1}(B)\supseteq A\,{\text{ if and only if }}\,f(A)\subseteq B$
$f(A)\supseteq f(X\setminus A)\,{\text{ if and only if }}\,f(A)=f(X)$	$f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\,{\text{ if and only if }}\,f^{-1}(B)=X$
$f(X\setminus A)\supseteq f(X)\setminus f(A)$	$f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)$
$f\left(A\cup f^{-1}(B)\right)\subseteq f(A)\cup B$	$f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)$
$f\left(A\cap f^{-1}(B)\right)=f(A)\cap B$	$f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)$

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:

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)	$f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)$
$f(A\setminus B)\supseteq f(A)\setminus f(B)$ (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 ofintersectionandunionwork 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, evenuncountably infinite.)

With respect to the algebra of subsets described above, the inverse image function is alattice homomorphism,while the image function is only asemilatticehomomorphism (that is, it does not always preserve intersections).

See also

• Bijection, injection and surjection– Properties of mathematical functions
• Fiber (mathematics)– Set of all points in a function's domain that all map to some single given point
• Image (category theory)– term in category theoryPages displaying wikidata descriptions as a fallback
• Kernel of a function– Equivalence relation expressing that two elements have the same image under a functionPages displaying short descriptions of redirect targets
• Set inversion– Mathematical problem of finding the set mapped by a specified function to a certain range

References

• Artin, Michael(1991).Algebra.Prentice Hall.ISBN81-203-0871-9.
• Blyth, T.S. (2005).Lattices and Ordered Algebraic Structures.Springer.ISBN1-85233-905-5..
• Dolecki, Szymon;Mynard, Frederic (2016).Convergence Foundations Of Topology.New Jersey: World Scientific Publishing Company.ISBN978-981-4571-52-4.OCLC945169917.
• Halmos, Paul R.(1960).Naive set theory.The University Series in Undergraduate Mathematics. van Nostrand Company.ISBN9780442030643.Zbl0087.04403.
• Kelley, John L. (1985).General Topology.Graduate Texts in Mathematics.Vol. 27 (2 ed.). Birkhäuser.ISBN978-0-387-90125-1.
• Munkres, James R.(2000).Topology(Second ed.).Upper Saddle River, NJ:Prentice Hall, Inc.ISBN978-0-13-181629-9.OCLC42683260.

This article incorporates material from Fibre onPlanetMath,which is licensed under theCreative Commons Attribution/Share-Alike License.