Editing Image (mathematics)

{{Short description|Set of the values of a function}}
[[File:FunctionMappingPersonToFavoriteFood.png|350px|thumb|right|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}.]]
{{other uses|Image (disambiguation)}}

In [[mathematics]], for a function <math>f: X \to Y</math>, the '''image''' of an input value <math>x</math> is the single output value produced by <math>f</math> when passed  <math>x</math>. The '''preimage''' of an output value <math>y</math> is the set of input values that produce <math>y</math>.

More generally, evaluating <math>f</math> at each [[Element (mathematics)|element]] of a given subset <math>A</math> of its [[Domain of a function|domain]] <math>X</math> produces a set, called the "'''image''' of <math>A</math> under (or through) <math>f</math>". Similarly, the '''inverse image''' (or '''preimage''') of a given subset <math>B</math> of the [[codomain]] <math>Y</math> is the set of all elements of <math>X</math> that map to a member of <math>B.</math>

The '''image''' of the function <math>f</math> is the set of all output values it may produce, that is, the image of <math>X</math>. The '''preimage''' of <math>f</math> is the preimage of the codomain <math>Y</math>. Because it always equals <math>X</math> (the domain of <math>f</math>), it is rarely used.

Image and inverse image may also be defined for general [[Binary relation#Operations|binary relations]], not just functions.

==Definition==
{{Group theory sidebar |Basics}}
[[File:ImagePreimageOfElement.png|300px|thumb|right|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y</math>. The image of element <math>x</math> is element <math>y</math>. The preimage of element <math>y</math> is the set {<math>x, x'</math>}. The preimage of element <math>y'</math> is <math>\varnothing</math>.]]
[[File:ImagePreimageofaSet.png|300px|thumb|right|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y</math>. The image of all elements in subset <math>A</math> is subset <math>B</math>. The preimage of <math>B</math> is subset <math>C</math>]]
[[File:Codomain2.SVG|thumb|upright=1.2|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y.</math> The yellow oval inside <math>Y</math> is the image of <math>f</math>. The preimage of <math>Y</math> is the entire domain <math>X</math>]]
The word "image" is used in three related ways. In these definitions, <math>f : X \to Y</math> is a [[Function (mathematics)|function]] from the [[Set (mathematics)|set]] <math>X</math> to the set <math>Y.</math>

===Image of an element===

If <math>x</math> is a member of <math>X,</math> then the image of <math>x</math> under <math>f,</math> denoted <math>f(x),</math> is the [[Value (mathematics)|value]] of <math>f</math> when applied to <math>x.</math> <math>f(x)</math> is alternatively known as the output of <math>f</math> for argument <math>x.</math> 

Given <math>y,</math> the function <math>f</math> is said to {{em|take the value <math>y</math>}} or {{em|take <math>y</math> as a value}} if there exists some <math>x</math> in the function's domain such that <math>f(x) = y.</math> 
Similarly, given a set <math>S,</math> <math>f</math> is said to {{em|take a value in <math>S</math>}} if there exists {{em|some}} <math>x</math> in the function's domain such that <math>f(x) \in S.</math> 
However, {{em|<math>f</math> takes [all] values in <math>S</math>}} and {{em|<math>f</math> is valued in <math>S</math>}} means that <math>f(x) \in S</math> for {{em|every}} point <math>x</math> in the domain of <math>f</math> .

===Image of a subset===

Throughout, let <math>f : X \to Y</math> be a function. 
The {{anchor|image of a set}}{{em|image}} under <math>f</math> of a subset <math>A</math> of <math>X</math> is the set of all <math>f(a)</math> for <math>a\in A.</math> It is denoted by <math>f[A],</math> or by <math>f(A)</math> when there is no risk of confusion. Using [[set-builder notation]], this definition can be written as<ref>{{Cite web|date=2019-11-05| title=5.4: Onto Functions and Images/Preimages of Sets| url=https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/5%3A_Functions/5.4%3A_Onto_Functions_and_Images%2F%2FPreimages_of_Sets| access-date=2020-08-28| website=Mathematics LibreTexts| language=en}}</ref><ref>{{cite book| author=Paul R. Halmos| title=Naive Set Theory| location=Princeton| publisher=Nostrand| year=1968 }} Here: Sect.8</ref>
<math display=block>f[A] = \{f(a) : a \in A\}.</math>

This induces a function <math>f[\,\cdot\,] : \mathcal P(X) \to \mathcal P(Y),</math> where <math>\mathcal P(S)</math> denotes the [[power set]] of a set <math>S;</math> that is the set of all [[subset]]s of <math>S.</math> See {{Section link||Notation}} below for more.

===Image of a function===

The ''image'' of a function is the image of its entire [[Domain of a function|domain]], also known as the [[Range of a function|range]] of the function.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Image|url=https://mathworld.wolfram.com/Image.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> This last usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>

===Generalization to binary relations===

If <math>R</math> is an arbitrary [[binary relation]] on <math>X \times Y,</math> then the set <math>\{ y \in Y : x R y \text{ for some } x \in X \}</math> is called the image, or the range, of <math>R.</math> Dually, the set <math>\{ x \in X : x R y \text{ for some } y \in Y \}</math> is called the domain of <math>R.</math>

==Inverse image==

{{Redirect|Preimage|the cryptographic attack on hash functions|preimage attack}}
Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by
<math display="block">f^{-1}[ B ] = \{ x \in X \,:\, f(x) \in B \}.</math>

Other notations include <math>f^{-1}(B)</math> and <math>f^{-}(B).</math>{{sfn|Dolecki|Mynard|2016|pp=4-5}} 
The inverse image of a [[Singleton (mathematics)|singleton set]], denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}(y),</math> is also called the [[Fiber (mathematics)|fiber]] or fiber over <math>y</math> or the [[level set]] of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>

For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for [[inverse function]], although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>

==<span id="Notation">Notation</span> for image and inverse image==
The traditional notations used in the previous section do not distinguish the original function <math>f : X \to Y</math> from the image-of-sets function <math>f : \mathcal{P}(X) \to \mathcal{P}(Y)</math>; 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{{sfn|Blyth|2005|p=5}} is to give explicit names for the image and preimage as functions between power sets:

===Arrow notation===

* <math>f^\rightarrow : \mathcal{P}(X) \to \mathcal{P}(Y)</math> with <math>f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}</math>
* <math>f^\leftarrow  : \mathcal{P}(Y) \to \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) \to \mathcal{P}(Y)</math> instead of <math>f^\rightarrow</math>
* <math>f^\star : \mathcal{P}(Y) \to \mathcal{P}(X)</math> instead of <math>f^\leftarrow</math>

===Other terminology===

* An alternative notation for <math>f[A]</math> used in [[mathematical logic]] and [[set theory]] is <math>f\,''A.</math><ref>{{cite book| title=Set Theory for the Mathematician|url=https://archive.org/details/settheoryformath0000rubi|url-access=registration|author=Jean E. Rubin |author-link= Jean E. Rubin |page=xix|year=1967 |publisher=Holden-Day |asin=B0006BQH7S}}</ref><ref>M. Randall Holmes: [https://web.archive.org/web/20180207010648/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 <math>f</math> as the range of <math>f,</math><ref>{{Cite book |last=Hoffman |first=Kenneth |title=Linear Algebra |publisher=Prentice-Hall |year=1971 |edition=2nd |pages=388 |language=en}}</ref> but this usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>

==Examples==
# <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>
    \left\{\begin{matrix}
      1 \mapsto a, \\
      2 \mapsto a, \\
      3 \mapsto c.
    \end{matrix}\right.
  </math>{{paragraph break}} The ''image'' of the set <math>\{ 2, 3 \}</math> under <math>f</math> is <math>f(\{ 2, 3 \}) = \{ a, c \}.</math> The ''image'' of the function <math>f</math> is <math>\{ a, c \}.</math> The ''preimage'' of <math>a</math> is <math>f^{-1}(\{ a \}) = \{ 1, 2 \}.</math> The ''preimage'' of <math>\{ a, b \}</math> is also <math>f^{-1}(\{ a, b \}) = \{ 1, 2 \}.</math> The ''preimage'' of <math>\{ b, d \}</math> under <math>f</math> is the [[empty set]] <math>\{ \ \} = \emptyset.</math>
# <math>f : \R \to \R</math> defined by <math>f(x) = x^2.</math>{{paragraph break}} The ''image'' of <math>\{ -2, 3 \}</math> under <math>f</math> is <math>f(\{ -2, 3 \}) = \{ 4, 9 \},</math> and the ''image'' of <math>f</math> is <math>\R^+</math> (the set of all positive real numbers and zero). The ''preimage'' of <math>\{ 4, 9 \}</math> under <math>f</math> is <math>f^{-1}(\{ 4, 9 \}) = \{ -3, -2, 2, 3 \}.</math> The ''preimage'' of set <math>N = \{ n \in \R : n < 0 \}</math> under <math>f</math> is the empty set, because the negative numbers do not have square roots in the set of reals.
# <math>f : \R^2 \to \R</math> defined by <math>f(x, y) = x^2 + y^2.</math>{{paragraph break}} The [[Fiber (mathematics)|''fibers'']] <math>f^{-1}(\{ a \})</math> are [[concentric circles]] about the [[Origin (mathematics)|origin]], the origin itself, and the [[empty set]] (respectively), depending on whether <math>a > 0, \ a = 0, \text{ or } \ a < 0</math> (respectively). (If <math>a \ge 0,</math> then the ''fiber'' <math>f^{-1}(\{ a \})</math> is the set of all <math>(x, y) \in \R^2</math> satisfying the equation <math>x^2 + y^2 = a,</math> that is, the origin-centered circle with radius <math>\sqrt{a}.</math>)
# If <math>M</math> is a [[manifold]] and <math>\pi : TM \to M</math> is the canonical [[Projection (mathematics)|projection]] from the [[tangent bundle]] <math>TM</math> to <math>M,</math> then the ''fibers'' of <math>\pi</math> are the [[tangent spaces]] <math>T_x(M) \text{ for } x \in M.</math> This is also an example of a [[fiber bundle]].
# A [[quotient group]] is a homomorphic ''image''.

== Properties ==

{{See also|List of set identities and relations#Functions and sets}}

{| class=wikitable style="float:right;"
|+
! Counter-examples based on the [[real number]]s <math>\R,</math><BR> <math>f : \R \to \R</math> defined by <math>x \mapsto x^2,</math><BR> showing that equality generally need<BR>not hold for some laws:
|-
|[[File:Image preimage conterexample intersection.gif|thumb|center|upright=1.2|Image showing non-equal sets: <math>f\left(A \cap B\right) \subsetneq f(A) \cap f(B).</math> The sets <math>A = [-4, 2]</math> and <math>B = [-2, 4]</math> are shown in {{color|blue|blue}} immediately below the <math>x</math>-axis while their intersection <math>A_3 = [-2, 2]</math> is shown in {{color|green|green}}.]]
|-
|[[File:Image preimage conterexample bf.gif|thumb|center|upright=1.2|<math>f\left(f^{-1}\left(B_3\right)\right) \subsetneq B_3.</math>]]
|-
|[[File:Image preimage conterexample fb.gif|thumb|center|upright=1.2|<math>f^{-1}\left(f\left(A_4\right)\right) \supsetneq A_4.</math>]]
|}

=== General ===

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

{| class="wikitable"
|-
! Image
! Preimage
|-
|<math>f(X) \subseteq Y</math>
|<math>f^{-1}(Y) = X</math>
|-
|<math>f\left(f^{-1}(Y)\right) = f(X)</math>
|<math>f^{-1}(f(X)) = X</math>
|-
|<math>f\left(f^{-1}(B)\right) \subseteq B</math><br>(equal if <math>B \subseteq f(X);</math> for instance, if <math>f</math> is surjective)<ref name="halmos-1960-p31">See {{harvnb|Halmos|1960|p=31}}</ref><ref name="munkres-2000-p19">See {{harvnb|Munkres|2000|p=19}}</ref>
|<math>f^{-1}(f(A)) \supseteq A</math><br>(equal if <math>f</math> is injective)<ref name="halmos-1960-p31"/><ref name="munkres-2000-p19" />
|-
|<math>f(f^{-1}(B)) = B \cap f(X)</math>
|<math>\left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)</math>
|-
|<math>f\left(f^{-1}(f(A))\right) = f(A)</math>
|<math>f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)</math>
|-
|<math>f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing</math>
|<math>f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)</math>
|-
|<math>f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B</math>
|<math>f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B</math>
|-
|<math>f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)</math>
|<math>f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X</math>
|-
|<math>f(X \setminus A) \supseteq f(X) \setminus f(A)</math>
|<math>f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)</math><ref name="halmos-1960-p31" />
|-
|<math>f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B</math><ref name="lee-2010-p388">See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.</ref>
|<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math><ref name="lee-2010-p388" />
|-
|<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math><ref name="lee-2010-p388" />
|<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math><ref name="lee-2010-p388" />
|}

Also:

* <math>f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing</math>

=== Multiple functions ===

For functions <math>f : X \to Y</math> and <math>g : Y \to Z</math> with subsets <math>A \subseteq X</math> and <math>C \subseteq Z,</math> the following properties hold:

* <math>(g \circ f)(A) = g(f(A))</math>
* <math>(g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))</math>

=== Multiple subsets of domain or codomain ===

For function <math>f : X \to Y</math> and subsets <math>A, B \subseteq X</math> and <math>S, T \subseteq Y,</math> the following properties hold:

{| class="wikitable"
|-
! Image
! Preimage
|-
|<math>A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)</math>
|<math>S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)</math>
|-
|<math>f(A \cup B) = f(A) \cup f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985">{{harvnb|Kelley|1985|p=[{{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}} 85]}}</ref>
|<math>f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)</math>
|-
|<math>f(A \cap B) \subseteq f(A) \cap f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985" /><br>(equal if <math>f</math> is injective<ref name="munkres-2000-p21">See {{harvnb|Munkres|2000|p=21}}</ref>)
|<math>f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)</math>
|-
|<math>f(A \setminus B) \supseteq f(A) \setminus f(B)</math><ref name="lee-2010-p388" /><br>(equal if <math>f</math> is injective<ref name="munkres-2000-p21" />)
|<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math><ref name="lee-2010-p388" />
|-
|<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math><br>(equal if <math>f</math> is injective)
|<math>f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)</math>
|-
|}

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

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

==See also==

* {{annotated link|Bijection, injection and surjection}}
* {{annotated link|Fiber (mathematics)}}
* {{annotated link|Image (category theory)}}
* {{annotated link|Kernel of a function}}
* {{annotated link|Set inversion}}

==Notes==

{{reflist}}
{{reflist|group=note}}

==References==

* {{Cite book|last=Artin|first=Michael|author-link=Michael Artin|title=Algebra|year=1991|publisher=Prentice Hall|isbn=81-203-0871-9}}
* {{cite book|first=T.S.|last=Blyth|title=Lattices and Ordered Algebraic Structures|publisher=Springer|year=2005|isbn=1-85233-905-5}}.
* {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|2016|p=}} -->
* {{cite book|last=Halmos|first=Paul R.|author-link=Paul Halmos|title=Naive set theory|url=https://archive.org/details/naivesettheory0000halm|url-access=registration|series=The University Series in Undergraduate Mathematics|publisher=van Nostrand Company|year=1960|isbn=9780442030643|zbl=0087.04403}}
* {{cite book|last1=Kelley|first1=John L.|title=General Topology|edition=2|series=[[Graduate Texts in Mathematics]]|volume=27|year=1985|publisher=Birkhäuser|isbn=978-0-387-90125-1}}
* {{Munkres Topology|edition=2}} <!-- {{sfn|Munkres|2000|p=}} -->
{{PlanetMath attribution|id=3276|title=Fibre}}

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