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Image (mathematics)

From Wikipedia, the free encyclopedia
Set of the values of a function
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, 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}.
For other uses, seeImage (disambiguation).

Inmathematics, for a functionf:X→Y{\displaystyle f:X\to Y}, theimage of an input valuex{\displaystyle x} is the single output value produced byf{\displaystyle f} when passedx{\displaystyle x}. Thepreimage of an output valuey{\displaystyle y} is the set of input values that producey{\displaystyle y}.

More generally, evaluatingf{\displaystyle f} at eachelement of a given subsetA{\displaystyle A} of itsdomainX{\displaystyle X} produces a set, called the "image ofA{\displaystyle A} under (or through)f{\displaystyle f}". Similarly, theinverse image (orpreimage) of a given subsetB{\displaystyle B} of thecodomainY{\displaystyle Y} is the set of all elements ofX{\displaystyle X} that map to a member ofB.{\displaystyle B.}

Theimage of the functionf{\displaystyle f} is the set of all output values it may produce, that is, the image ofX{\displaystyle X}. Thepreimage off{\displaystyle f} is the preimage of the codomainY{\displaystyle Y}. Because it always equalsX{\displaystyle X} (the domain off{\displaystyle f}), it is rarely used.

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

Definition

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Algebraic structure →Group theory
Group theory
f{\displaystyle f} is a function from domainX{\displaystyle X} to codomainY{\displaystyle Y}. The image of elementx{\displaystyle x} is elementy{\displaystyle y}. The preimage of elementy{\displaystyle y} is the set {x,x′{\displaystyle x,x'}}. The preimage of elementy′{\displaystyle y'} is∅{\displaystyle \varnothing }.
f{\displaystyle f} is a function from domainX{\displaystyle X} to codomainY{\displaystyle Y}. The image of all elements in subsetA{\displaystyle A} is subsetB{\displaystyle B}. The preimage ofB{\displaystyle B} is subsetC{\displaystyle C}
f{\displaystyle f} is a function from domainX{\displaystyle X} to codomainY.{\displaystyle Y.} The yellow oval insideY{\displaystyle Y} is the image off{\displaystyle f}. The preimage ofY{\displaystyle Y} is the entire domainX{\displaystyle X}

The word "image" is used in three related ways. In these definitions,f:X→Y{\displaystyle f:X\to Y} is afunction from thesetX{\displaystyle X} to the setY.{\displaystyle Y.}

Image of an element

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Ifx{\displaystyle x} is a member ofX,{\displaystyle X,} then the image ofx{\displaystyle x} underf,{\displaystyle f,} denotedf(x),{\displaystyle f(x),} is thevalue off{\displaystyle f} when applied tox.{\displaystyle x.}f(x){\displaystyle f(x)} is alternatively known as the output off{\displaystyle f} for argumentx.{\displaystyle x.}

Giveny,{\displaystyle y,} the functionf{\displaystyle f} is said totake the valuey{\displaystyle y} ortakey{\displaystyle y} as a value if there exists somex{\displaystyle x} in the function's domain such thatf(x)=y.{\displaystyle f(x)=y.} Similarly, given a setS,{\displaystyle S,}f{\displaystyle f} is said totake a value inS{\displaystyle S} if there existssomex{\displaystyle x} in the function's domain such thatf(x)∈S.{\displaystyle f(x)\in S.} However,f{\displaystyle f} takes [all] values inS{\displaystyle S} andf{\displaystyle f} is valued inS{\displaystyle S} means thatf(x)∈S{\displaystyle f(x)\in S} forevery pointx{\displaystyle x} in the domain off{\displaystyle f} .

Image of a subset

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Throughout, letf:X→Y{\displaystyle f:X\to Y} be a function. Theimage underf{\displaystyle f} of a subsetA{\displaystyle A} ofX{\displaystyle X} is the set of allf(a){\displaystyle f(a)} fora∈A.{\displaystyle a\in A.} It is denoted byf[A],{\displaystyle f[A],} or byf(A){\displaystyle f(A)} when there is no risk of confusion. Usingset-builder notation, this definition can be written as[1][2]f[A]={f(a):a∈A}.{\displaystyle f[A]=\{f(a):a\in A\}.}

This induces a functionf[⋅]:P(X)→P(Y),{\displaystyle f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),} whereP(S){\displaystyle {\mathcal {P}}(S)} denotes thepower set of a setS;{\displaystyle S;} that is the set of allsubsets ofS.{\displaystyle S.} See§ Notation below for more.

Image of a function

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Theimage of a function is the image of its entiredomain, also known as therange of the function.[3] This last usage should be avoided because the word "range" is also commonly used to mean thecodomain off.{\displaystyle f.}

Generalization to binary relations

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IfR{\displaystyle R} is an arbitrarybinary relation onX×Y,{\displaystyle X\times Y,} then the set{y∈Y:xRy for some x∈X}{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called[by whom?] the image, or the range, ofR.{\displaystyle R.} Dually, the set{x∈X:xRy for some y∈Y}{\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called[by whom?] the domain ofR.{\displaystyle R.}

Inverse image

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

Letf{\displaystyle f} be a function fromX{\displaystyle X} toY.{\displaystyle Y.} Thepreimage orinverse image of a setB⊆Y{\displaystyle B\subseteq Y} underf,{\displaystyle f,} denoted byf−1[B],{\displaystyle f^{-1}[B],} is the subset ofX{\displaystyle X} defined byf−1[B]={x∈X:f(x)∈B}.{\displaystyle f^{-1}[B]=\{x\in X\,:\,f(x)\in B\}.}

Other notations includef−1(B){\displaystyle f^{-1}(B)} andf−(B).{\displaystyle f^{-}(B).}[4] The inverse image of asingleton set, denoted byf−1[{y}]{\displaystyle f^{-1}[\{y\}]} or byf−1(y),{\displaystyle f^{-1}(y),} is also called thefiber or fiber overy{\displaystyle y} or thelevel set ofy.{\displaystyle y.} The set of all the fibers over the elements ofY{\displaystyle Y} is afamily of sets indexed byY.{\displaystyle Y.}

For example, for the functionf(x)=x2,{\displaystyle f(x)=x^{2},} the inverse image of{4}{\displaystyle \{4\}} would be{−2,2}.{\displaystyle \{-2,2\}.} Again, if there is no risk of confusion,f−1[B]{\displaystyle f^{-1}[B]} can be denoted byf−1(B),{\displaystyle f^{-1}(B),} andf−1{\displaystyle f^{-1}} can also be thought of as a function from the power set ofY{\displaystyle Y} to the power set ofX.{\displaystyle X.} The notationf−1{\displaystyle f^{-1}} should not be confused with that forinverse function, although it coincides with the usual one for bijections in that the inverse image ofB{\displaystyle B} underf{\displaystyle f} is the image ofB{\displaystyle B} underf−1.{\displaystyle f^{-1}.}

Notation for image and inverse image

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The traditional notations used in the previous section do not distinguish the original functionf:X→Y{\displaystyle f:X\to Y} from the image-of-sets functionf:P(X)→P(Y){\displaystyle 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 alternative[5] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

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Star notation

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Other terminology

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Examples

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  1. f:{1,2,3}→{a,b,c,d}{\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}} defined by{1↦a,2↦a,3↦c.{\displaystyle \left\{{\begin{matrix}1\mapsto a,\\2\mapsto a,\\3\mapsto c.\end{matrix}}\right.}
    Theimage of the set{2,3}{\displaystyle \{2,3\}} underf{\displaystyle f} isf({2,3})={a,c}.{\displaystyle f(\{2,3\})=\{a,c\}.} Theimage of the functionf{\displaystyle f} is{a,c}.{\displaystyle \{a,c\}.} Thepreimage ofa{\displaystyle a} isf−1({a})={1,2}.{\displaystyle f^{-1}(\{a\})=\{1,2\}.} Thepreimage of{a,b}{\displaystyle \{a,b\}} is alsof−1({a,b})={1,2}.{\displaystyle f^{-1}(\{a,b\})=\{1,2\}.} Thepreimage of{b,d}{\displaystyle \{b,d\}} underf{\displaystyle f} is theempty set{ }=∅.{\displaystyle \{\ \}=\emptyset .}
  2. f:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } defined byf(x)=x2.{\displaystyle f(x)=x^{2}.}
    Theimage of{−2,3}{\displaystyle \{-2,3\}} underf{\displaystyle f} isf({−2,3})={4,9},{\displaystyle f(\{-2,3\})=\{4,9\},} and theimage off{\displaystyle f} isR+{\displaystyle \mathbb {R} ^{+}} (the set of allpositive real numbers and zero). Thepreimage of{4,9}{\displaystyle \{4,9\}} underf{\displaystyle f} isf−1({4,9})={−3,−2,2,3}.{\displaystyle f^{-1}(\{4,9\})=\{-3,-2,2,3\}.} Thepreimage of setN={n∈R:n<0}{\displaystyle N=\{n\in \mathbb {R} :n<0\}} underf{\displaystyle f} is the empty set, because the negative numbers do not have square roots in the set of reals.
  3. f:R2→R{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } defined byf(x,y)=x2+y2.{\displaystyle f(x,y)=x^{2}+y^{2}.}
    Thefibersf−1({a}){\displaystyle f^{-1}(\{a\})} areconcentric circles about theorigin, the origin itself, and theempty set (respectively), depending on whethera>0, a=0, or  a<0{\displaystyle a>0,\ a=0,{\text{ or }}\ a<0} (respectively). (Ifa≥0,{\displaystyle a\geq 0,} then thefiberf−1({a}){\displaystyle f^{-1}(\{a\})} is the set of all(x,y)∈R2{\displaystyle (x,y)\in \mathbb {R} ^{2}} satisfying the equationx2+y2=a,{\displaystyle x^{2}+y^{2}=a,} that is, the origin-centered circle with radiusa.{\displaystyle {\sqrt {a}}.})
  4. IfM{\displaystyle M} is amanifold andπ:TM→M{\displaystyle \pi :TM\to M} is the canonicalprojection from thetangent bundleTM{\displaystyle TM} toM,{\displaystyle M,} then thefibers ofπ{\displaystyle \pi } are thetangent spacesTx(M) for x∈M.{\displaystyle T_{x}(M){\text{ for }}x\in M.} This is also an example of afiber bundle.
  5. Aquotient group is a homomorphicimage.

Properties

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See also:List of set identities and relations § Functions and sets
Counter-examples based on thereal numbersR,{\displaystyle \mathbb {R} ,}
f:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } defined byx↦x2,{\displaystyle x\mapsto x^{2},}
showing that equality generally need
not hold for some laws:
Image showing non-equal sets:f(A∩B)⊊f(A)∩f(B).{\displaystyle f\left(A\cap B\right)\subsetneq f(A)\cap f(B).} The setsA=[−4,2]{\displaystyle A=[-4,2]} andB=[−2,4]{\displaystyle B=[-2,4]} are shown inblue immediately below thex{\displaystyle x}-axis while their intersectionA3=[−2,2]{\displaystyle A_{3}=[-2,2]} is shown ingreen.
f(f−1(B3))⊊B3.{\displaystyle f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.}
f−1(f(A4))⊋A4.{\displaystyle f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.}

General

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For every functionf:X→Y{\displaystyle f:X\to Y} and all subsetsA⊆X{\displaystyle A\subseteq X} andB⊆Y,{\displaystyle B\subseteq Y,} the following properties hold:

ImagePreimage
f(X)⊆Y{\displaystyle f(X)\subseteq Y}f−1(Y)=X{\displaystyle f^{-1}(Y)=X}
f(f−1(Y))=f(X){\displaystyle f\left(f^{-1}(Y)\right)=f(X)}f−1(f(X))=X{\displaystyle f^{-1}(f(X))=X}
f(f−1(B))⊆B{\displaystyle f\left(f^{-1}(B)\right)\subseteq B}
(equal ifB⊆f(X);{\displaystyle B\subseteq f(X);} for instance, iff{\displaystyle f} is surjective)[9][10]		f−1(f(A))⊇A{\displaystyle f^{-1}(f(A))\supseteq A}
(equal iff{\displaystyle f} is injective)[9][10]
f(f−1(B))=B∩f(X){\displaystyle f(f^{-1}(B))=B\cap f(X)}(f|A)−1(B)=A∩f−1(B){\displaystyle \left(f\vert _{A}\right)^{-1}(B)=A\cap f^{-1}(B)}
f(f−1(f(A)))=f(A){\displaystyle f\left(f^{-1}(f(A))\right)=f(A)}f−1(f(f−1(B)))=f−1(B){\displaystyle f^{-1}\left(f\left(f^{-1}(B)\right)\right)=f^{-1}(B)}
f(A)=∅ if and only if A=∅{\displaystyle f(A)=\varnothing \,{\text{ if and only if }}\,A=\varnothing }f−1(B)=∅ if and only if B⊆Y∖f(X){\displaystyle f^{-1}(B)=\varnothing \,{\text{ if and only if }}\,B\subseteq Y\setminus f(X)}
f(A)⊇B if and only if  there exists C⊆A such that f(C)=B{\displaystyle f(A)\supseteq B\,{\text{ if and only if }}{\text{ there exists }}C\subseteq A{\text{ such that }}f(C)=B}f−1(B)⊇A if and only if f(A)⊆B{\displaystyle f^{-1}(B)\supseteq A\,{\text{ if and only if }}\,f(A)\subseteq B}
f(A)⊇f(X∖A) if and only if f(A)=f(X){\displaystyle f(A)\supseteq f(X\setminus A)\,{\text{ if and only if }}\,f(A)=f(X)}f−1(B)⊇f−1(Y∖B) if and only if f−1(B)=X{\displaystyle f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\,{\text{ if and only if }}\,f^{-1}(B)=X}
f(X∖A)⊇f(X)∖f(A){\displaystyle f(X\setminus A)\supseteq f(X)\setminus f(A)}f−1(Y∖B)=X∖f−1(B){\displaystyle f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)}[9]
f(A∪f−1(B))⊆f(A)∪B{\displaystyle f\left(A\cup f^{-1}(B)\right)\subseteq f(A)\cup B}[11]f−1(f(A)∪B)⊇A∪f−1(B){\displaystyle f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)}[11]
f(A∩f−1(B))=f(A)∩B{\displaystyle f\left(A\cap f^{-1}(B)\right)=f(A)\cap B}[11]f−1(f(A)∩B)⊇A∩f−1(B){\displaystyle f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)}[11]

Also:

Multiple functions

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For functionsf:X→Y{\displaystyle f:X\to Y} andg:Y→Z{\displaystyle g:Y\to Z} with subsetsA⊆X{\displaystyle A\subseteq X} andC⊆Z,{\displaystyle C\subseteq Z,} the following properties hold:

Multiple subsets of domain or codomain

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For functionf:X→Y{\displaystyle f:X\to Y} and subsetsA,B⊆X{\displaystyle A,B\subseteq X} andS,T⊆Y,{\displaystyle S,T\subseteq Y,} the following properties hold:

ImagePreimage
A⊆B implies f(A)⊆f(B){\displaystyle A\subseteq B\,{\text{ implies }}\,f(A)\subseteq f(B)}S⊆T implies f−1(S)⊆f−1(T){\displaystyle S\subseteq T\,{\text{ implies }}\,f^{-1}(S)\subseteq f^{-1}(T)}
f(A∪B)=f(A)∪f(B){\displaystyle f(A\cup B)=f(A)\cup f(B)}[11][12]f−1(S∪T)=f−1(S)∪f−1(T){\displaystyle f^{-1}(S\cup T)=f^{-1}(S)\cup f^{-1}(T)}
f(A∩B)⊆f(A)∩f(B){\displaystyle f(A\cap B)\subseteq f(A)\cap f(B)}[11][12]
(equal iff{\displaystyle f} is injective[13])		f−1(S∩T)=f−1(S)∩f−1(T){\displaystyle f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)}
f(A∖B)⊇f(A)∖f(B){\displaystyle f(A\setminus B)\supseteq f(A)\setminus f(B)}[11]
(equal iff{\displaystyle f} is injective[13])		f−1(S∖T)=f−1(S)∖f−1(T){\displaystyle f^{-1}(S\setminus T)=f^{-1}(S)\setminus f^{-1}(T)}[11]
f(A△B)⊇f(A)△f(B){\displaystyle f\left(A\triangle B\right)\supseteq f(A)\triangle f(B)}
(equal iff{\displaystyle f} is injective)		f−1(S△T)=f−1(S)△f−1(T){\displaystyle f^{-1}\left(S\triangle T\right)=f^{-1}(S)\triangle f^{-1}(T)}

The results relating images and preimages to the (Boolean) algebra ofintersection andunion work for any collection of subsets, not just for pairs of subsets:

(Here,S{\displaystyle 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 asemilattice homomorphism (that is, it does not always preserve intersections).

See also

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Notes

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  1. ^"5.4: Onto Functions and Images/Preimages of Sets".Mathematics LibreTexts. 2019-11-05. Retrieved2020-08-28.
  2. ^Paul R. Halmos (1968).Naive Set Theory. Princeton: Nostrand. Here: Sect.8
  3. ^Weisstein, Eric W."Image".mathworld.wolfram.com. Retrieved2020-08-28.
  4. ^Dolecki & Mynard 2016, pp. 4–5.
  5. ^Blyth 2005, p. 5.
  6. ^Jean E. Rubin (1967).Set Theory for the Mathematician. Holden-Day. p. xix.ASIN B0006BQH7S.
  7. ^M. Randall Holmes:Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
  8. ^Hoffman, Kenneth (1971).Linear Algebra (2nd ed.). Prentice-Hall. p. 388.
  9. ^abcSeeHalmos 1960, p. 31
  10. ^abSeeMunkres 2000, p. 19
  11. ^abcdefghSee p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
  12. ^abKelley 1985, p. 85
  13. ^abSeeMunkres 2000, p. 21

References

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

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