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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>]]
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| | In mathematics, the image of a function is the set of all output values it may produce. | | In mathematics, the image of a function is the set of all output values it may produce. |
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| − | More generally, evaluating a given function {\displaystyle f}f 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. | + | 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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| | Image and inverse image may also be defined for general binary relations, not just functions. | | Image and inverse image may also be defined for general 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 from the 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 is the power set of <math>X</math> (the set of all subsets of <math>X</math>), and whose codomain is the power set of <math>Y.</math>
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| − | ===Image of a function===
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| − | The ''image'' of a function is the image of its entire domain, also known as the range of the function. 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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| − | 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 set, denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}[y],</math> is also called the 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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| − | ==Notation 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>
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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'' <math>f^{-1}(\{ a \})</math> are concentric circles about the 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 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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| − | ! Counter-examples based on the real numbers <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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| − | |[[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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| − | ! Image
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| − | ! Preimage
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| − | |<math>f(X) \subseteq Y</math>
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| − | |<math>f^{-1}(Y) = X</math>
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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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| − | |<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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| − | |<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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| − | |<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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| − | |<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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| − | |<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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| − | ! Image
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| − | ! Preimage
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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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| − | |<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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| − | The results relating images and preimages to the (Boolean) algebra of intersection and 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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| − | ==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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