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| [[File:Codomain2.SVG|thumb|upright=1.5|''f'' is a function from domain ''X'' to codomain ''Y''. The smaller oval inside ''Y'' is the image of ''f''.]]
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| {{Group theory sidebar |Basics}}
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| In [[mathematics]], an '''image''' is the [[subset]] of a function's [[codomain]] which is the output of the function on a subset of its [[Domain of a function|domain]]. Precisely evaluating the function at each element of a subset X of the domain produces a set called the image of X ''under or through'' the function. The '''inverse image''' or '''preimage''' of a particular subset ''S'' of the [[codomain]] of a function is the set of all elements of the domain that map to the members of ''S''.
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| Image and inverse image may also be defined for general [[binary relation#Operations on binary relations|binary relations]], not just functions.
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| ==Definition==
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| The word "image" is used in three related ways. In these definitions, ''f'' : ''X'' → ''Y'' is a [[function (mathematics)|function]] from the [[Set (mathematics)|set]] ''X'' to the set ''Y''.
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| ===Image of an element===
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| If ''x'' is a member of ''X'', then ''f''(''x'') = ''y'' (the [[value (mathematics)|value]] of ''f'' when applied to ''x'') is the image of ''x'' under ''f''. ''y'' is alternatively known as the output of ''f'' for argument ''x''.
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| ===Image of a subset===
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| The image of a subset ''A'' ⊆ ''X'' under ''f'' is the subset ''f''<nowiki>[</nowiki>''A''<nowiki>]</nowiki> ⊆ ''Y'' defined by (in [[set-builder notation]]):
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| :<math>f[A] = \{ \, y \in Y \, | \, y = f(x) \text{ for some } x \in A \, \}</math> | |
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| When there is no risk of confusion, ''f''<nowiki>[</nowiki>''A''<nowiki>]</nowiki> is simply written as ''f''(''A''). This convention is a common one; the intended meaning must be inferred from the context. This makes the image of ''f'' a function whose [[domain of a function|domain]] is the [[power set]] of ''X'' (the set of all [[subset]]s of ''X''), and whose [[codomain]] is the power set of ''Y''. See [[#Notation|Notation]] below.
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| ===Image of a function===
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| The image ''f''<nowiki>[</nowiki>''X''<nowiki>]</nowiki> of the entire [[domain of a function|domain]] ''X'' of ''f'' is called simply the image of ''f''.
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| ==Inverse image==
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| {{Redirect|Preimage|the cryptographic attack on hash functions|preimage attack}}
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| Let ''f'' be a function from ''X'' to ''Y''. The preimage or inverse image of a set ''B'' ⊆ ''Y'' under ''f'' is the subset of ''X'' defined by
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| :<math>f^{-1}[ B ] = \{ \, x \in X \, | \, f(x) \in B \}</math>
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| The inverse image of a [[singleton (mathematics)|singleton]], denoted by ''f''<sup> −1</sup><nowiki>[</nowiki>{''y''}<nowiki>]</nowiki> or by ''f''<sup> −1</sup><nowiki>[</nowiki>''y''<nowiki>]</nowiki>, is also called the [[fiber (mathematics)|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''.
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| For example, for the function ''f''(''x'') = ''x''<sup>2</sup>, the inverse image of {4} would be {-2,2}. Again, if there is no risk of confusion, we may denote ''f''<sup> −1</sup><nowiki>[</nowiki>''B''<nowiki>]</nowiki> by ''f''<sup> −1</sup>(''B''), and think of ''f''<sup> −1</sup> as a function from the power set of ''Y'' to the power set of ''X''. The notation ''f''<sup> −1</sup> should not be confused with that for [[inverse function]]. The two coincide only if ''f'' is a [[bijection]].
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| ==<span id="Notation">Notation</span> for image and inverse image==
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| The traditional notations used in the previous section can be confusing. An alternative<ref>Blyth 2005, p. 5</ref> is to give explicit names for the image and preimage as functions between powersets: | |
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| ===Arrow notation===
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| * <math>f^\rightarrow:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)</math> with <math>f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}</math>
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| * <math>f^\leftarrow:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)</math> with <math>f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}</math>
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| ===Star notation===
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| * <math>f_\star:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)</math> instead of <math>f^\rightarrow</math>
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| * <math>f^\star:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)</math> instead of <math>f^\leftarrow</math>
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| ===Other terminology===
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| * An alternative notation for ''f''<nowiki>[</nowiki>''A''<nowiki>]</nowiki> used in [[mathematical logic]] and [[set theory]] is ''f'' "''A''.<ref>{{cite book| title=Set Theory for the Mathematician | author=Jean E. Rubin |page=xix | year=1967 |publisher=Holden-Day |asin=B0006BQH7S}}</ref>
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| * 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''.
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| ==Examples==
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| 1. ''f'': {1,2,3} → {''a,b,c,d''} defined by <math>f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ a, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.</math>
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| 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''<sup> −1</sup>({''a''}) = {1,2}. The ''preimage'' of {''a,b''} is also {1,2}. The preimage of {''b'',''d''} is the [[empty set]] {}.
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| 2. ''f'': '''R''' → '''R''' defined by ''f''(''x'') = ''x''<sup>2</sup>.
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| The ''image'' of {-2,3} under ''f'' is ''f''({-2,3}) = {4,9}, and the ''image'' of ''f'' is '''R<sup>+</sup>'''. The ''preimage'' of {4,9} under ''f'' is ''f''<sup> −1</sup>({4,9}) = {-3,-2,2,3}. The preimage of set ''N'' = {''n'' ∈ '''R''' | ''n'' < 0} under ''f'' is the empty set, because the negative numbers do not have square roots in the set of reals.
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| 3. ''f'': '''R'''<sup>2</sup> → '''R''' defined by ''f''(''x'', ''y'') = ''x''<sup>2</sup> + ''y''<sup>2</sup>.
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| The ''fibres'' ''f''<sup> −1</sup>({''a''}) are [[concentric circles]] about the [[origin (mathematics)|origin]], the origin itself, and the [[empty set]], depending on whether ''a''>0, ''a''=0, or ''a''<0, respectively.
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| 4. If ''M'' is a [[manifold]] and ''π'' :''TM''→''M'' is the canonical [[projection (mathematics)|projection]] from the [[tangent bundle]] ''TM'' to ''M'', then the ''fibres'' of ''π'' are the [[tangent spaces]] ''T''<sub>''x''</sub>(''M'') for ''x''∈''M''. This is also an example of a [[fiber bundle]].
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| ==Consequences==
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| Given a function ''f'' : ''X'' → ''Y'', for all subsets ''A'', ''A''<sub>1</sub>, and ''A''<sub>2</sub> of ''X'' and all subsets ''B'', ''B''<sub>1</sub>, and ''B''<sub>2</sub> of ''Y'' we have:
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| *''f''(''A''<sub>1</sub> ∪ ''A''<sub>2</sub>) = ''f''(''A''<sub>1</sub>) ∪ ''f''(''A''<sub>2</sub>)<ref name="kelley-1985">Kelley (1985), {{Google books quote|id=-goleb9Ov3oC|page=85|text=The image of the union of a family of subsets of X is the union of the images, but, in general, the image of the intersection is not the intersection of the images|p. 85}}</ref>
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| *''f''(''A''<sub>1</sub> ∩ ''A''<sub>2</sub>) ⊆ ''f''(''A''<sub>1</sub>) ∩ ''f''(''A''<sub>2</sub>)<ref name="kelley-1985"/>
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| *''f''<sup> −1</sup>(''B''<sub>1</sub> ∪ ''B''<sub>2</sub>) = ''f''<sup> −1</sup>(''B''<sub>1</sub>) ∪ ''f''<sup> −1</sup>(''B''<sub>2</sub>)
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| *''f''<sup> −1</sup>(''B''<sub>1</sub> ∩ ''B''<sub>2</sub>) = ''f''<sup> −1</sup>(''B''<sub>1</sub>) ∩ ''f''<sup> −1</sup>(''B''<sub>2</sub>)
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| *''f''(A) ⊆ ''B'' ⇔ ''A'' ⊆ ''f''<sup> −1</sup>(''B'')
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| *''f''(''f''<sup> −1</sup>(''B'')) ⊆ ''B''<ref>Equality holds if ''B'' is a subset of Im(''f'') or, in particular, if ''f'' is surjective. See Munkres, J.. Topology (2000), p. 19.</ref>
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| *''f''<sup> −1</sup>(''f''(''A'')) ⊇ ''A''<ref>Equality holds if ''f'' is injective. See Munkres, J.. Topology (2000), p. 19.</ref>
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| *''A''<sub>1</sub> ⊆ ''A''<sub>2</sub> ⇒ ''f''(''A''<sub>1</sub>) ⊆ ''f''(''A''<sub>2</sub>)
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| *''B''<sub>1</sub> ⊆ ''B''<sub>2</sub> ⇒ ''f''<sup> −1</sup>(''B''<sub>1</sub>) ⊆ ''f''<sup> −1</sup>(''B''<sub>2</sub>)
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| *''f''<sup> −1</sup>(''B''<sup>C</sup>) = (''f''<sup> −1</sup>(''B''))<sup>C</sup>
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| *(''f'' |<sub>''A''</sub>)<sup>−1</sup>(''B'') = ''A'' ∩ ''f''<sup> −1</sup>(''B'').
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| The results relating images and preimages to the ([[Boolean algebra (structure)|Boolean]]) algebra of [[intersection (set theory)|intersection]] and [[union (set theory)|union]] work for any collection of subsets, not just for pairs of subsets:
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| *<math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f(A_s)</math>
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| *<math>f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f(A_s)</math>
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| *<math>f^{-1}\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f^{-1}(A_s)</math>
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| *<math>f^{-1}\left(\bigcap_{s\in S}A_s\right) = \bigcap_{s\in S} f^{-1}(A_s)</math>
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| (Here, ''S'' can be infinite, even [[uncountably infinite]].)
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| With respect to the algebra of subsets, by the above we see that the inverse image function is a [[lattice homomorphism]] while the image function is only a [[semilattice]] homomorphism (it does not always preserve intersections).
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| ==See also==
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| *[[Range (mathematics)]]
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| *[[Bijection, injection and surjection]]
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| *[[Kernel of a function]]
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| *[[Image (category theory)]]
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| * [[Set inversion]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Cite book |authorlink=Michael Artin | last= Artin | first= Michael | title= Algebra | edition=| year=1991 | publisher=Prentice Hall| isbn= 81-203-0871-9 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * T.S. Blyth, ''Lattices and Ordered Algebraic Structures'', Springer, 2005, ISBN 1-85233-905-5.
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| * {{cite book |last1=Munkres |first1=James R. |authorlink1= |last2= |first2= |authorlink2= |title=Topology |url= |edition=2 |series= |volume= |year=2000 |publisher=Prentice Hall |location= |isbn=978-0-13-181629-9 |id= }}
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| * {{cite book |last1=Kelley |first1=John L. |authorlink1= |last2= |first2= |authorlink2= |title=General Topology |url= |edition=2 |series=Graduate texts in mathematics |volume=27 |year=1985 |publisher=Birkhäuser |location= |isbn=978-0-387-90125-1 |id= }}
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| {{PlanetMath attribution|id=3276|title=Fibre}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Isomorphism theorems]]
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