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| − | [[File:Codomain2.SVG|thumb|upright=1.5|<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>]]
| + | In mathematics, the image of a function is the set of all output values it may produce. |
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| − | In [[mathematics]], the '''image''' of a [[Function (mathematics)|function]] is the set of all output values it may produce.
| + | More generally, evaluating a given function <math>f</math> at each element of a given subset {\displaystyle A}A of its domain produces a set, called the "image of {\displaystyle A}A under (or through) {\displaystyle f}f". Similarly, the inverse image (or preimage) of a given subset {\displaystyle B}B of the codomain of {\displaystyle f,}f, is the set of all elements of the domain that map to the members of {\displaystyle B.}B. |
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| − | More generally, evaluating a given function <math>f</math> at each [[Element (mathematics)|element]] of a given subset <math>A</math> of its [[Domain of a function|domain]] 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]] of <math>f,</math> is the set of all elements of the domain that map to the members of <math>B.</math>
| + | Image and inverse image may also be defined for general binary relations, not just functions. |
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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, <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>
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| − | ===Image of an element===
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| − | 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>
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| − | 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>
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| − | 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>
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| − | 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 <math>f</math>'s domain.
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| − | ===Image of a subset===
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| − | The image of a subset <math>A \subseteq X</math> under <math>f,</math> denoted <math>f[A],</math> is the subset of <math>Y</math> which can be defined using [[set-builder notation]] as follows:<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>
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| − | <math display=block>f[A] = \{ f(x) : x \in A \}</math>
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| − | When there is no risk of confusion, <math>f[A]</math> is simply written as <math>f(A).</math> This convention is a common one; the intended meaning must be inferred from the context. This makes <math>f[\,\cdot\,]</math> a function whose [[Domain of a function|domain]] is the [[power set]] of <math>X</math> (the set of all [[subset]]s of <math>X</math>), and whose [[codomain]] is the power set of <math>Y.</math> See {{Section link||Notation}} below for more.
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| − | ===Image of a function===
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| − | 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 usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>
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| − | ===Generalization to binary relations===
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| − | 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>
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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 <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
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| − | <math display=block>f^{-1}[ B ] = \{ x \in X \,|\, f(x) \in B \}.</math>
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| − | Other notations include <math>f^{-1}(B)</math><ref>{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-08-28|website=Math Vault|language=en-US}}</ref> and <math>f^{-}(B).</math>{{sfn|Dolecki|Mynard|2016|pp=4-5}}
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| − | 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>
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| − | 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>
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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{{sfn|Blyth|2005|p=5}} is to give explicit names for the image and preimage as functions between power sets:
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| − | ===Arrow notation===
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| − | * <math>f^\rightarrow : \mathcal{P}(X) \to \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) \to \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) \to \mathcal{P}(Y)</math> instead of <math>f^\rightarrow</math>
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| − | * <math>f^\star : \mathcal{P}(Y) \to \mathcal{P}(X)</math> instead of <math>f^\leftarrow</math>
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| − | ===Other terminology===
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| − | * 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>
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| − | * Some texts refer to the image of <math>f</math> as the range of <math>f,</math> but this usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>
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| − | ==Examples==
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| − | # <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>
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| − | f(x) = \left\{\begin{matrix}
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| − | a, & \mbox{if }x=1 \\
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| − | a, & \mbox{if }x=2 \\
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| − | c, & \mbox{if }x=3.
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| − | \end{matrix}\right.
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| − | </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}(\{ 1, 2 \}) = \{ 1, 2 \}.</math> The preimage of <math>\{ b, d \},</math> is the [[empty set]] <math>\{ \, \} = \varnothing.</math>
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| − | # <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^{-1}(\{ -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.
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| − | # <math>f : \R^2 \to \R</math> defined by <math>f(x, y) = x^2 + y^2.</math>{{paragraph break}} The [[Fiber (mathematics)|''fiber'']] <math>f^{-1}(\{ a \})</math> are [[concentric circles]] about the [[Origin (mathematics)|origin]], the origin itself, and the [[empty set]], depending on whether <math>a > 0, a = 0, \text{ or } a < 0,</math> respectively. (if <math>a > 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 of the origin-concentric ring <math>x^2 + y^2 = a.</math>)
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| − | # 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]].
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| − | # A [[quotient group]] is a homomorphic image.
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| − | == Properties ==
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| − | {{See also|List of set identities and relations#Functions and sets}}
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| − | {| class=wikitable style="float:right;"
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| − | |+
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| − | ! 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:
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| − | |[[File:Image preimage conterexample intersection.gif|thumb|center|upright=1.7|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}}.]]
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| − | |[[File:Image preimage conterexample bf.gif|thumb|center|upright=1.7|<math>f\left(f^{-1}\left(B_3\right)\right) \subsetneq B_3.</math>]]
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| − | |-
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| − | |[[File:Image preimage conterexample fb.gif|thumb|center|upright=1.7|<math>f^{-1}\left(f\left(A_4\right)\right) \supsetneq A_4.</math>]]
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| − | |}
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| − | === General ===
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| − | 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:
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| − | {| class="wikitable"
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| − | |-
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| − | ! Image
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| − | ! Preimage
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| − | |-
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| − | |<math>f(X) \subseteq Y</math>
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| − | |<math>f^{-1}(Y) = X</math>
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| − | |-
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| − | |<math>f\left(f^{-1}(Y)\right) = f(X)</math>
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| − | |<math>f^{-1}(f(X)) = X</math>
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| − | |<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-p39">See {{harvnb|Halmos|1960|p=39}}</ref><ref name="munkres-2000-p19">See {{harvnb|Munkres|2000|p=19}}</ref>
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| − | |<math>f^{-1}(f(A)) \supseteq A</math><br>(equal if <math>f</math> is injective)<ref name="halmos-1960-p39"/><ref name="munkres-2000-p19" />
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| − | |-
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| − | |<math>f(f^{-1}(B)) = B \cap f(X)</math>
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| − | |<math>\left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)</math>
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| − | |-
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| − | |<math>f\left(f^{-1}(f(A))\right) = f(A)</math>
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| − | |<math>f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)</math>
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| − | |<math>f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing</math>
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| − | |<math>f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)</math>
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| − | |-
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| − | |<math>f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B</math>
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| − | |<math>f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B</math>
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| − | |<math>f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)</math>
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| − | |<math>f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X</math>
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| − | |-
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| − | |<math>f(X \setminus A) \supseteq f(X) \setminus f(A)</math>
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| − | |<math>f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)</math><ref name="halmos-1960-p39" />
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| − | |<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>
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| − | |<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math><ref name="lee-2010-p388" />
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| − | |-
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| − | |<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math><ref name="lee-2010-p388" />
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| − | |<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math><ref name="lee-2010-p388" />
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| − | |}
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| − | Also:
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| − | * <math>f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing</math>
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| − | === Multiple functions ===
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| − | 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:
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| − | * <math>(g \circ f)(A) = g(f(A))</math>
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| − | * <math>(g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))</math>
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| − | | |
| − | === Multiple subsets of domain or codomain ===
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| − | 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:
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| − | | |
| − | {| class="wikitable"
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| − | |-
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| − | ! Image
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| − | ! Preimage
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| − | |-
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| − | |<math>A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)</math>
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| − | |<math>S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)</math>
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| − | |-
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| − | |<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>
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| − | |<math>f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)</math>
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| − | |-
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| − | |<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>)
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| − | |<math>f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)</math>
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| − | |-
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| − | |<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" />)
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| − | |<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math><ref name="lee-2010-p388" />
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| − | |-
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| − | |<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math><br>(equal if <math>f</math> is injective)
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| − | |<math>f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)</math>
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| − | |-
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| − | |}
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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\left(A_s\right)</math>
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| − | * <math>f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f\left(A_s\right)</math>
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| − | * <math>f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)</math>
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| − | * <math>f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)</math>
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| − | (Here, <math>S</math> can be infinite, even [[uncountably infinite]].)
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| − | 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).
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| − | ==See also==
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| − | | |
| − | * {{annotated link|Bijection, injection and surjection}}
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| − | * {{annotated link|Image (category theory)}}
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| − | * {{annotated link|Kernel of a function}}
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| − | * {{annotated link|Set inversion}}
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| − | ==Notes==
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| − | | |
| − | {{reflist}}
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| − | {{reflist|group=note}}
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| − | | |
| − | ==References==
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| − | | |
| − | * {{Cite book|last=Artin|first=Michael|author-link=Michael Artin|title=Algebra|year=1991|publisher=Prentice Hall|isbn=81-203-0871-9}}
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| − | * {{cite book|first=T.S.|last=Blyth|title=Lattices and Ordered Algebraic Structures|publisher=Springer|year=2005|isbn=1-85233-905-5}}.
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| − | * {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|2016|p=}} -->
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| − | * {{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}}
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| − | * {{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}}
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| − | * {{Munkres Topology|edition=2}} <!-- {{sfn|Munkres|2000|p=}} -->
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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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