View source for Image (mathematics)

{{short description|The set of all values of a function}}
{{other uses|Image (disambiguation)}}
[[File:Codomain2.SVG|thumb|upright=1.5|''f'' is a function from domain ''X'' to codomain ''Y''. The yellow oval inside ''Y'' is the image of ''f''.]]
{{Group theory sidebar |Basics}}

In [[mathematics]], the '''image''' of a [[Function (mathematics)|function]] is the set of all output values it may take. 

More generally, evaluating a given function ''f'' at each element of a given subset ''A'' of its [[Domain of a function|domain]] produces a set called the "'''image''' of ''A'' under (or through) ''f'' ". 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 relation#Operations on binary relations|binary relations]], not just functions.

==Definition==
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''.

===Image of an element===
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''.

===Image of a subset===
The image of a subset ''A'' ⊆ ''X'' under ''f'' is the subset ''f''<nowiki>[</nowiki>''A''<nowiki>]</nowiki> ⊆ ''Y'' defined by (using [[set-builder notation]]):

:<math>f[A] = \{ f(x) \mid x \in A \}</math>

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 ''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.

===Image of a function===
The ''image'' of a function is the image of its entire [[domain of a function|domain]].

===Generalization to binary relations===
If ''R'' is an arbitrary [[binary relation]] on ''X''[[cartesian product|×]]''Y'', the set { y∈''Y'' | ''xRy'' for some ''x''∈''X'' } is called the image, or the range, of ''R''. Dually, the set { ''x''∈''X'' | ''xRy'' for some y∈''Y'' } is called the domain of ''R''.

==Inverse image==
{{Redirect|Preimage|the cryptographic attack on hash functions|preimage attack}}
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

:<math>f^{-1}[ B ] = \{ x \in X \, | \, f(x) \in B \}.</math>

The inverse image of a [[singleton (mathematics)|singleton]], denoted by ''f''<sup>&nbsp;−1</sup><nowiki>[</nowiki>{''y''}<nowiki>]</nowiki> or by   ''f''<sup>&nbsp;−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''.

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, denote ''f''<sup>&nbsp;−1</sup><nowiki>[</nowiki>''B''<nowiki>]</nowiki> by ''f''<sup>&nbsp;−1</sup>(''B''), and think of ''f''<sup>&nbsp;−1</sup> as a function from the power set of ''Y'' to the power set of ''X''. The notation ''f''<sup>&nbsp;−1</sup> should not be confused with that for [[inverse function]]. The notation coincides with the usual one, though, for bijections, in the sense that the inverse image of ''B'' under ''f'' is the image of ''B'' under ''f''<sup>&nbsp;−1</sup>.

==<span id="Notation">Notation</span> for image and inverse image==
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:

===Arrow notation===
* <math>f^\rightarrow:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)</math> with <math>f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}</math>
* <math>f^\leftarrow:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)</math> with <math>f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}</math>

===Star notation===
* <math>f_\star:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)</math> instead of <math>f^\rightarrow</math>
* <math>f^\star:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)</math> instead of <math>f^\leftarrow</math>

===Other terminology===
* An alternative notation for ''f''<nowiki>[</nowiki>''A''<nowiki>]</nowiki> used in [[mathematical logic]] and [[set theory]] is ''f''&nbsp;"''A''.<ref>{{cite book| title=Set Theory for the Mathematician | url=https://archive.org/details/settheoryformath0000rubi | url-access=registration | author=Jean E. Rubin |authorlink= Jean E. Rubin |page=xix | year=1967 |publisher=Holden-Day |asin=B0006BQH7S}}</ref><ref>M. Randall Holmes: [https://pdfs.semanticscholar.org/d8d8/5cdd3eb2fd9406d13b5c04d55708068031ef.pdf Inhomogeneity of the urelements in the usual models of NFU], December 29, 2005, on: Semantic Scholar, p. 2</ref>
* 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'': {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>{{paragraph break}} 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>&nbsp;−1</sup>({''a''}) = {1, 2}. The ''preimage'' of {''a, b''} is also {1, 2}.  The preimage of {''b'', ''d''} is the [[empty set]] {}.
# ''f'': '''R''' → '''R''' defined by ''f''(''x'') = ''x''<sup>2</sup>.{{paragraph break}} 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>&nbsp;−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.
# ''f'': '''R'''<sup>2</sup> → '''R''' defined by ''f''(''x'', ''y'') = ''x''<sup>2</sup> + ''y''<sup>2</sup>.{{paragraph break}} The [[Fiber (mathematics)|''fibres'']] ''f''<sup>&nbsp;−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.
# 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]].
# A quotient group is a homomorphic image.

==Properties==

{| class=wikitable style="float:right;"
|+
! Counter-examples based on<BR> ''f'':[[real number|ℝ]]→ℝ, ''x''↦''x''<sup>2</sup>, showing<BR>that equality generally need<BR>not hold for some laws:
|-
| [[File:Image preimage conterexample intersection.gif|thumb|''f''(''A''<sub>1</sub>∩''A''<sub>2</sub>) ⊊ ''f''(''A''<sub>1</sub>) ∩ ''f''(''A''<sub>2</sub>)]]
|-
| [[File:Image preimage conterexample bf.gif|thumb|''f''(''f''<sup>−1</sup>(''B''<sub>3</sub>)) ⊊ ''B''<sub>3</sub>]]
|-
| [[File:Image preimage conterexample fb.gif|thumb|''f''<sup>−1</sup>(''f''(''A''<sub>4</sub>)) ⊋ ''A''<sub>4</sub>]]
|}

For every function ''f'' : ''X'' → ''Y'', 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'', the following properties hold:
* ''f''(''A''<sub>1</sub>&nbsp;∪&nbsp;''A''<sub>2</sub>)&nbsp;= ''f''(''A''<sub>1</sub>)&nbsp;∪&nbsp;''f''(''A''<sub>2</sub>)<ref name="kelley-1985">Kelley (1985), [{{Google books|plainurl=y|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>
* ''f''(''A''<sub>1</sub>&nbsp;∩&nbsp;''A''<sub>2</sub>)&nbsp;⊆ ''f''(''A''<sub>1</sub>)&nbsp;∩&nbsp;''f''(''A''<sub>2</sub>)<ref name="kelley-1985"/>
* ''f''(''A''&nbsp;∩&nbsp;''f''<sup>&nbsp;−1</sup>(''B'')) = ''f''(''A'')&nbsp;∩&nbsp;''B''
* ''f''<sup>&nbsp;−1</sup>(''B''<sub>1</sub>&nbsp;∪&nbsp;''B''<sub>2</sub>)&nbsp;= ''f''<sup>&nbsp;−1</sup>(''B''<sub>1</sub>)&nbsp;∪&nbsp;''f''<sup>&nbsp;−1</sup>(''B''<sub>2</sub>)
* ''f''<sup>&nbsp;−1</sup>(''B''<sub>1</sub>&nbsp;∩&nbsp;''B''<sub>2</sub>)&nbsp;= ''f''<sup>&nbsp;−1</sup>(''B''<sub>1</sub>)&nbsp;∩&nbsp;''f''<sup>&nbsp;−1</sup>(''B''<sub>2</sub>)
* ''f''(''A'') = ∅ ⇔ ''A'' = ∅
* ''f''<sup>&nbsp;−1</sup>(''B'') = ∅ ⇔ ''B''&nbsp;⊆&nbsp;(''f''(''X''))<sup>C</sup>
* ''f''(''A'')&nbsp;∩&nbsp;''B'' = ∅ ⇔ ''A''&nbsp;∩&nbsp;''f''<sup>&nbsp;−1</sup>(''B'') = ∅
* ''f''(''A'')&nbsp;⊆&nbsp;''B'' ⇔  ''A''&nbsp;⊆&nbsp;    ''f''<sup>&nbsp;−1</sup>(''B'')
* ''B''&nbsp;⊆&nbsp;''f''(''A'') ⇔ <math>\exists</math>''C''&nbsp;⊆&nbsp;''A'' (''f''(''C'') = ''B'')
* ''f''(''f''<sup>&nbsp;−1</sup>(''B''))&nbsp;⊆&nbsp;''B''<ref>Equality holds if ''B'' is a subset of ''f''(''X'') or, in particular, if ''f'' is surjective. See Munkres, J.. Topology (2000), p. 19.</ref>
* ''f''<sup>&nbsp;−1</sup>(''f''(''A''))&nbsp;⊇&nbsp;''A''<ref>Equality holds if  ''f'' is injective. See Munkres, J.. Topology (2000), p. 19.</ref>
* ''f''(''f''<sup>&nbsp;−1</sup>(''B'')) = B&nbsp;∩&nbsp;''f''(''X'')
* ''f''<sup>&nbsp;−1</sup>(''f''(''X'')) = ''X''
* ''A''<sub>1</sub> ⊆ ''A''<sub>2</sub> ⇒ ''f''(''A''<sub>1</sub>) ⊆ ''f''(''A''<sub>2</sub>)
* ''B''<sub>1</sub> ⊆ ''B''<sub>2</sub> ⇒ ''f''<sup>&nbsp;−1</sup>(''B''<sub>1</sub>) ⊆ ''f''<sup>&nbsp;−1</sup>(''B''<sub>2</sub>)
* ''f''<sup>&nbsp;−1</sup>(''B''<sup>C</sup>) = (''f''<sup>&nbsp;−1</sup>(''B''))<sup>C</sup>
* (''f''&nbsp;|<sub>''A''</sub>)<sup>−1</sup>(''B'') = ''A'' ∩ ''f''<sup>&nbsp;−1</sup>(''B'').

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:
* <math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f(A_s)</math>
* <math>f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f(A_s)</math>
* <math>f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}(B_s)</math>
* <math>f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}(B_s)</math>
(Here, ''S'' can be infinite, even [[uncountably infinite]].)

With respect to the algebra of subsets, by the above the inverse image function is a [[lattice homomorphism]] while the image function is only a [[semilattice]] homomorphism (it does not always preserve intersections).

==See also==
* [[Range (mathematics)]]
* [[Bijection, injection and surjection]]
* [[Kernel of a function]]
* [[Image (category theory)]]
* [[Set inversion]]

==Notes==
{{reflist}}

==References==
* {{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}}}}
* T.S. Blyth, ''Lattices and Ordered Algebraic Structures'', Springer, 2005, {{ISBN|1-85233-905-5}}.
* {{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= }}
* {{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= }}
{{PlanetMath attribution|id=3276|title=Fibre}}

[[Category:Basic concepts in set theory]]
[[Category:Isomorphism theorems]]