Difference between revisions of "Functions:Forward Image"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
 
(28 intermediate revisions by one other user not shown)
Line 1: Line 1:
−
[[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.
  
−
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 <math>A</math> of its 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>.
  
−
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.
−
 
 
−
Image and inverse image may also be defined for general [[Binary relation#Operations on binary relations|binary relations]], not just functions.
 
  
 
==Definition==
 
==Definition==
−
 
+
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>.
−
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===
 
===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 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>.
  
−
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 "take the value <math>y</math>" or "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 "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain.
−
 
 
−
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 <math>f</math>'s domain.
 
  
 
===Image of a subset===
 
===Image of a subset===
 +
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:
  
−
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>
+
: <math>f[A]=\{f(x):x\in A\}</math>
−
<math display=block>f[A] = \{ f(x) : x \in A \}</math>
 
  
−
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.
+
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>.
  
 
===Image of a function===
 
===Image of a function===
−
 
+
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 {\displaystyle f.}f.
−
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>
 
  
 
===Generalization to binary relations===
 
===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:xRy \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:xRy{\text{ for some }}y\in Y\}</math> is called the domain of <math>R</math>.
  
−
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 of a function==
−
 
+
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>f^{-1}[ B ] = \{ x \in X \,|\, f(x) \in B \}.</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><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}}
 
−
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 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:
 
−
 
 
−
===Arrow notation===
 
  
−
* <math>f^\rightarrow : \mathcal{P}(X) \to \mathcal{P}(Y)</math> with <math>f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}</math>
+
Other notations include <math>f^{-1}(B)</math> and <math>f^{-}(B).</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>
+
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>
  
−
===Star notation===
+
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>
−
 
 
−
* <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> but this usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>
 
  
 
==Examples==
 
==Examples==
  
−
# <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>
+
1. <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>
 
     f(x) = \left\{\begin{matrix}
 
     f(x) = \left\{\begin{matrix}
 
       a, & \mbox{if }x=1 \\
 
       a, & \mbox{if }x=1 \\
Line 71: Line 42:
 
       c, & \mbox{if }x=3.
 
       c, & \mbox{if }x=3.
 
     \end{matrix}\right.
 
     \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}(\{ 1, 2 \}) = \{ 1, 2 \}.</math>  The preimage of <math>\{ b, d \},</math> is the [[empty set]] <math>\{ \, \} = \varnothing.</math>
+
   </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^{-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.
+
: 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>
−
# <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>)
+
 
−
# 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]].
+
2. <math>f : \R \to \R</math> defined by <math>f(x) = x^2.</math>
−
# A [[quotient group]] is a homomorphic image.
+
: 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.
 +
 
 +
3. <math>f : \R^2 \to \R</math> defined by <math>f(x, y) = x^2 + y^2.</math>
 +
: 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>)
 +
 
 +
4. 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.
 +
 
 +
5. A quotient group is a homomorphic image.
  
 
== Properties ==
 
== Properties ==
  
−
{{See also|List of set identities and relations#Functions and sets}}
+
[[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>]]
  
−
{| class=wikitable style="float:right;"
+
[[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>]]
−
|+
 
−
! 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=1.7|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}}.]]
 
−
|-
 
−
|[[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>]]
 
−
|-
 
−
|[[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>]]
 
−
|}
 
  
 
=== General ===
 
=== General ===
Line 107: Line 76:
 
|<math>f^{-1}(f(X)) = 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-p39">See {{harvnb|Halmos|1960|p=39}}</ref><ref name="munkres-2000-p19">See {{harvnb|Munkres|2000|p=19}}</ref>
+
|<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)
−
|<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" />
 
 
|-
 
|-
 
|<math>f(f^{-1}(B)) = B \cap f(X)</math>
 
|<math>f(f^{-1}(B)) = B \cap f(X)</math>
Line 126: Line 94:
 
|-
 
|-
 
|<math>f(X \setminus A) \supseteq f(X) \setminus f(A)</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-p39" />
+
|<math>f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)</math>
 
|-
 
|-
−
|<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\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B</math>
−
|<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math><ref name="lee-2010-p388" />
+
|<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math>
 
|-
 
|-
−
|<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math><ref name="lee-2010-p388" />
+
|<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math>
−
|<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math><ref name="lee-2010-p388" />
+
|<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math>
 
|}
 
|}
  
Line 158: Line 126:
 
|<math>S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)</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(A \cup B) = f(A) \cup f(B)</math>
 
|<math>f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)</math>
 
|<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(A \cap B) \subseteq f(A) \cap f(B)</math>
 
|<math>f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)</math>
 
|<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(A \setminus B) \supseteq f(A) \setminus f(B)</math>
−
|<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math><ref name="lee-2010-p388" />
+
|<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math>
 
|-
 
|-
 
|<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math><br>(equal if <math>f</math> is injective)
 
|<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math><br>(equal if <math>f</math> is injective)
Line 172: Line 140:
 
|}
 
|}
  
−
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:
+
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:
  
 
* <math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)</math>
 
* <math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)</math>
Line 178: Line 146:
 
* <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(\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>
 
* <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]].)
+
(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|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}}
+
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).
−
* {{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]]
+
== Licensing ==
−
[[Category:Isomorphism theorems]]
+
Content obtained and/or adapted from:
 +
* [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics), Wikipedia] under a CC BY-SA license

Latest revision as of 15:11, 6 November 2021

In mathematics, the image of a function is the set of all output values it may produce.

More generally, evaluating a given function at each element of a given subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of its domain produces a set, called the "image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} under (or through) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ". Similarly, the inverse image (or preimage) of a given subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of the codomain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , is the set of all elements of the domain that map to the members of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} .

Image and inverse image may also be defined for general binary relations, not just functions.

Definition

The word "image" is used in three related ways. In these definitions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to\ Y} is a function from the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .

Image of an element

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a member of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , then the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , is the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} when applied to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is alternatively known as the output of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} for argument Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is said to "take the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} " or "take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} as a value" if there exists some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the function's domain such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=y} . Similarly, given a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is said to "take a value in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} " if there exists some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the function's domain such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\in S} . However, "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} takes all values in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} " and "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is valued in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} " means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\in S} for every point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} 's domain.

Image of a subset

The image of a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subseteq X} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f[A]} , is the subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} which can be defined using set-builder notation as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f[A]=\{f(x):x\in A\}}

When there is no risk of confusion, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f[A]} is simply written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A)} . This convention is a common one; the intended meaning must be inferred from the context. This makes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f[\,\cdot \,]} a function whose domain is the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} (the set of all subsets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ), and whose codomain is the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .

Image of a function

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 {\displaystyle f.}f.

Generalization to binary relations

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is an arbitrary binary relation on , then the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{y\in Y:xRy \text{ for some } x\in X\}} is called the image, or the range, of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . Dually, the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

Inverse image of a function

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be a function from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y.} The preimage or inverse image of a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \subseteq Y} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f,} denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[B],} is the subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[ B ] = \{ x \in X \,|\, f(x) \in B \}.}

Other notations include Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-}(B).} The inverse image of a singleton set, denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[\{ y \}]} or by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[y],} is also called the fiber or fiber over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} or the level set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y.} The set of all the fibers over the elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} is a family of sets indexed by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y.}

For example, for the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2,} the inverse image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 4 \}} would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ -2, 2 \}.} Again, if there is no risk of confusion, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}[B]} can be denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B),} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} can also be thought of as a function from the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} to the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X.} The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}.}

Examples

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}} defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{\begin{matrix} a, & \mbox{if }x=1 \\ a, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right. }

The image of the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 2, 3 \}} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\{ 2, 3 \}) = \{ a, c \}.} The image of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ a, c \}.} The preimage of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\{ a \}) = \{ 1, 2 \}.} The preimage of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ a, b \}} is also Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\{ 1, 2 \}) = \{ 1, 2 \}.} The preimage of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ b, d \},} is the empty set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ \, \} = \varnothing.}

2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \R \to \R} defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2.}

The image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ -2, 3 \}} under is and the image of is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^+} (the set of all positive real numbers and zero). The preimage of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 4, 9 \}} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\{ 4, 9 \}) = \{ -3, -2, 2, 3 \}.} The preimage of set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \{ n \in \R : n < 0 \}} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the empty set, because the negative numbers do not have square roots in the set of reals.

3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \R^2 \to \R} defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x, y) = x^2 + y^2.}

The fiber Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\{ a \})} are concentric circles]] about the origin, the origin itself, and the empty set, depending on whether Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 0, a = 0, \text{ or } a < 0,} respectively. (if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 0,} then the fiber Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\{ a \})} is the set of all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x, y) \in \R^2} satisfying the equation of the origin-concentric ring Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + y^2 = a.} )

4. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a manifold and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi : TM \to M} is the canonical projection from the tangent bundle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TM} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M,} then the fibers of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} are the tangent spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_x(M) \text{ for } x \in M.} This is also an example of a fiber bundle.

5. A quotient group is a homomorphic image.

Properties

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(f^{-1}\left(B_3\right)\right) \subsetneq B_3.}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}\left(f\left(A_4\right)\right) \supsetneq A_4.}

General

For every function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : X \to Y} and all subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \subseteq Y,} the following properties hold:

Image Preimage
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X) \subseteq Y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(Y) = X}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(f^{-1}(Y)\right) = f(X)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(f(X)) = X}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(f^{-1}(B)\right) \subseteq B}
(equal if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \subseteq f(X);} for instance, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is surjective)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(f^{-1}(B)) = B \cap f(X)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(f^{-1}(f(A))\right) = f(A)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X \setminus A) \supseteq f(X) \setminus f(A)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(A \cap f^{-1}(B)\right) = f(A) \cap B} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)}

Also:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing}

Multiple functions

For functions and with subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \subseteq Z,} the following properties hold:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g \circ f)(A) = g(f(A))}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))}

Multiple subsets of domain or codomain

For function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : X \to Y} and subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \subseteq X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S, T \subseteq Y,} the following properties hold:

Image Preimage
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A \cup B) = f(A) \cup f(B)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A \cap B) \subseteq f(A) \cap f(B)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A \setminus B) \supseteq f(A) \setminus f(B)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)}
(equal if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is injective) 		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)}

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:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f\left(A_s\right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)}

(Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} 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).

Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Functions:Forward_Image&oldid=3534"