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<h2 id="definition">Definition</h2> <p>The word "image" is used in three related ways. In these definitions, f : X → Y {\displaystyle f:X\to Y} is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> from the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> X {\displaystyle X} to the set Y . {\displaystyle Y.} </p> <h3>Image of an element</h3> <p>If x {\displaystyle x} is a member of X , {\displaystyle X,} then the image of x {\displaystyle x} under f , {\displaystyle f,} denoted f ( x ) , {\displaystyle f(x),} is the <a href="/facts/Value_(mathematics)/mGYQu2xt">value</a> of f {\displaystyle f} when applied to x . {\displaystyle x.} f ( x ) {\displaystyle f(x)} is alternatively known as the output of f {\displaystyle f} for argument x . {\displaystyle x.} </p><p>Given y , {\displaystyle y,} the function f {\displaystyle f} is said to take the value y {\displaystyle y} or take y {\displaystyle y} as a value if there exists some x {\displaystyle x} in the function's domain such that f ( x ) = y . {\displaystyle f(x)=y.} Similarly, given a set S , {\displaystyle S,} f {\displaystyle f} is said to take a value in S {\displaystyle S} if there exists some x {\displaystyle x} in the function's domain such that f ( x ) ∈ S . {\displaystyle f(x)\in S.} However, f {\displaystyle f} takes [all] values in S {\displaystyle S} and f {\displaystyle f} is valued in S {\displaystyle S} means that f ( x ) ∈ S {\displaystyle f(x)\in S} for every point x {\displaystyle x} in the domain of f {\displaystyle f} . </p> <h3>Image of a subset</h3> <p>Throughout, let f : X → Y {\displaystyle f:X\to Y} be a function. The image under f {\displaystyle f} of a subset A {\displaystyle A} of X {\displaystyle X} is the set of all f ( a ) {\displaystyle f(a)} for a ∈ A . {\displaystyle a\in A.} It is denoted by f [ A ] , {\displaystyle f[A],} or by f ( A ) {\displaystyle f(A)} when there is no risk of confusion. Using <a href="/facts/Set-builder_notation/0nVHQF6E">set-builder notation</a>, this definition can be written as<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> f [ A ] = { f ( a ) : a ∈ A } . {\displaystyle f[A]=\{f(a):a\in A\}.} </p><p>This induces a function f [ ⋅ ] : P ( X ) → P ( Y ) , {\displaystyle f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),} where P ( S ) {\displaystyle {\mathcal {P}}(S)} denotes the <a href="/facts/Power_set/FVuf8PDD">power set</a> of a set S ; {\displaystyle S;} that is the set of all <a href="/facts/Subset/SJGERmbA">subsets</a> of S . {\displaystyle S.} See § Notation below for more. </p> <h3>Image of a function</h3> <p>The <i>image</i> of a function is the image of its entire <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, also known as the <a href="/facts/Range_of_a_function/66CeTV6u">range</a> of the function.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> This last usage should be avoided because the word "range" is also commonly used to mean the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of f . {\displaystyle f.} </p> <h3>Generalization to binary relations</h3> <p>If R {\displaystyle R} is an arbitrary <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a> on X × Y , {\displaystyle X\times Y,} then the set { y ∈ Y : x R y for some x ∈ X } {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of R . {\displaystyle R.} Dually, the set { x ∈ X : x R y for some y ∈ Y } {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}} is called the domain of R . {\displaystyle R.} </p> <h2 id="inverse-image">Inverse image</h2> <p class="note">"Preimage" redirects here. For the cryptographic attack on hash functions, see <a href="/facts/Preimage_attack/dDktzjys">preimage attack</a>.</p> <p>Let f {\displaystyle f} be a function from X {\displaystyle X} to Y . {\displaystyle Y.} The preimage or inverse image of a set B ⊆ Y {\displaystyle B\subseteq Y} under f , {\displaystyle f,} denoted by f − 1 [ B ] , {\displaystyle f^{-1}[B],} is the subset of X {\displaystyle X} defined by f − 1 [ B ] = { x ∈ X : f ( x ) ∈ B } . {\displaystyle f^{-1}[B]=\{x\in X\,:\,f(x)\in B\}.} </p><p>Other notations include f − 1 ( B ) {\displaystyle f^{-1}(B)} and f − ( B ) . {\displaystyle f^{-}(B).} <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The inverse image of a <a href="/facts/Singleton_(mathematics)/LtNVfujT">singleton set</a>, denoted by f − 1 [ { y } ] {\displaystyle f^{-1}[\{y\}]} or by f − 1 ( y ) , {\displaystyle f^{-1}(y),} is also called the <a href="/facts/Fiber_(mathematics)/rL3pwFwa">fiber</a> or fiber over y {\displaystyle y} or the <a href="/facts/Level_set/JUmNbdNh">level set</a> of y . {\displaystyle y.} The set of all the fibers over the elements of Y {\displaystyle Y} is a family of sets indexed by Y . {\displaystyle Y.} </p><p>For example, for the function f ( x ) = x 2 , {\displaystyle f(x)=x^{2},} the inverse image of { 4 } {\displaystyle \{4\}} would be { − 2 , 2 } . {\displaystyle \{-2,2\}.} Again, if there is no risk of confusion, f − 1 [ B ] {\displaystyle f^{-1}[B]} can be denoted by f − 1 ( B ) , {\displaystyle f^{-1}(B),} and f − 1 {\displaystyle f^{-1}} can also be thought of as a function from the power set of Y {\displaystyle Y} to the power set of X . {\displaystyle X.} The notation f − 1 {\displaystyle f^{-1}} should not be confused with that for <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a>, although it coincides with the usual one for bijections in that the inverse image of B {\displaystyle B} under f {\displaystyle f} is the image of B {\displaystyle B} under f − 1 . {\displaystyle f^{-1}.} </p> <h2 id="notation-for-image-and-inverse-image">Notation for image and inverse image</h2> <p>The traditional notations used in the previous section do not distinguish the original function f : X → Y {\displaystyle f:X\to Y} from the image-of-sets function f : P ( X ) → P ( Y ) {\displaystyle f:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)} ; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> is to give explicit names for the image and preimage as functions between power sets: </p> <h3>Arrow notation</h3> <ul><li> f → : P ( X ) → P ( Y ) {\displaystyle f^{\rightarrow }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)} with f → ( A ) = { f ( a ) | a ∈ A } {\displaystyle f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}} </li> <li> f ← : P ( Y ) → P ( X ) {\displaystyle f^{\leftarrow }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)} with f ← ( B ) = { a ∈ X | f ( a ) ∈ B } {\displaystyle f^{\leftarrow }(B)=\{a\in X\;|\;f(a)\in B\}} </li></ul> <h3>Star notation</h3> <ul><li> f ⋆ : P ( X ) → P ( Y ) {\displaystyle f_{\star }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)} instead of f → {\displaystyle f^{\rightarrow }} </li> <li> f ⋆ : P ( Y ) → P ( X ) {\displaystyle f^{\star }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)} instead of f ← {\displaystyle f^{\leftarrow }} </li></ul> <h3>Other terminology</h3> <ul><li>An alternative notation for f [ A ] {\displaystyle f[A]} used in <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and <a href="/facts/Set_theory/1ptfsvUP">set theory</a> is f ″ A . {\displaystyle f\,''A.} <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li>Some texts refer to the image of f {\displaystyle f} as the range of f , {\displaystyle f,} <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> but this usage should be avoided because the word "range" is also commonly used to mean the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of f . {\displaystyle f.} </li></ul> <h2 id="examples">Examples</h2> <ol><li> f : { 1 , 2 , 3 } → { a , b , c , d } {\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}} defined by { 1 ↦ a , 2 ↦ a , 3 ↦ c . {\displaystyle \left\{{\begin{matrix}1\mapsto a,\\2\mapsto a,\\3\mapsto c.\end{matrix}}\right.} The <i>image</i> of the set { 2 , 3 } {\displaystyle \{2,3\}} under f {\displaystyle f} is f ( { 2 , 3 } ) = { a , c } . {\displaystyle f(\{2,3\})=\{a,c\}.} The <i>image</i> of the function f {\displaystyle f} is { a , c } . {\displaystyle \{a,c\}.} The <i>preimage</i> of a {\displaystyle a} is f − 1 ( { a } ) = { 1 , 2 } . {\displaystyle f^{-1}(\{a\})=\{1,2\}.} The <i>preimage</i> of { a , b } {\displaystyle \{a,b\}} is also f − 1 ( { a , b } ) = { 1 , 2 } . {\displaystyle f^{-1}(\{a,b\})=\{1,2\}.} The <i>preimage</i> of { b , d } {\displaystyle \{b,d\}} under f {\displaystyle f} is the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> { } = ∅ . {\displaystyle \{\ \}=\emptyset .} </li> <li> f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = x 2 . {\displaystyle f(x)=x^{2}.} The <i>image</i> of { − 2 , 3 } {\displaystyle \{-2,3\}} under f {\displaystyle f} is f ( { − 2 , 3 } ) = { 4 , 9 } , {\displaystyle f(\{-2,3\})=\{4,9\},} and the <i>image</i> of f {\displaystyle f} is R + {\displaystyle \mathbb {R} ^{+}} (the set of all positive real numbers and zero). The <i>preimage</i> of { 4 , 9 } {\displaystyle \{4,9\}} under f {\displaystyle f} is f − 1 ( { 4 , 9 } ) = { − 3 , − 2 , 2 , 3 } . {\displaystyle f^{-1}(\{4,9\})=\{-3,-2,2,3\}.} The <i>preimage</i> of set N = { n ∈ R : n < 0 } {\displaystyle N=\{n\in \mathbb {R} :n<0\}} under f {\displaystyle f} is the empty set, because the negative numbers do not have square roots in the set of reals.</li> <li> f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } defined by f ( x , y ) = x 2 + y 2 . {\displaystyle f(x,y)=x^{2}+y^{2}.} The <a href="/facts/Fiber_(mathematics)/rL3pwFwa"><i>fibers</i></a> f − 1 ( { a } ) {\displaystyle f^{-1}(\{a\})} are <a href="/facts/Concentric_circles/xR1I5N8y">concentric circles</a> about the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>, the origin itself, and the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> (respectively), depending on whether a > 0 , a = 0 , or a < 0 {\displaystyle a>0,\ a=0,{\text{ or }}\ a<0} (respectively). (If a ≥ 0 , {\displaystyle a\geq 0,} then the <i>fiber</i> f − 1 ( { a } ) {\displaystyle f^{-1}(\{a\})} is the set of all ( x , y ) ∈ R 2 {\displaystyle (x,y)\in \mathbb {R} ^{2}} satisfying the equation x 2 + y 2 = a , {\displaystyle x^{2}+y^{2}=a,} that is, the origin-centered circle with radius a . {\displaystyle {\sqrt {a}}.} )</li> <li>If M {\displaystyle M} is a <a href="/facts/Manifold/rXPERJtu">manifold</a> and π : T M → M {\displaystyle \pi :TM\to M} is the canonical <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a> from the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> T M {\displaystyle TM} to M , {\displaystyle M,} then the <i>fibers</i> of π {\displaystyle \pi } are the <a href="/facts/Tangent_spaces/crvpbSeG">tangent spaces</a> T x ( M ) for x ∈ M . {\displaystyle T_{x}(M){\text{ for }}x\in M.} This is also an example of a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a>.</li> <li>A <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> is a homomorphic <i>image</i>.</li></ol> <h2 id="properties">Properties</h2> <p class="note">See also: <a href="/facts/List_of_set_identities_and_relations/yekH29va">List of set identities and relations § Functions and sets</a></p> <table><tbody><tr><th>Counter-examples based on the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> R , {\displaystyle \mathbb {R} ,} f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by x ↦ x 2 , {\displaystyle x\mapsto x^{2},} showing that equality generally neednot hold for some laws:</th></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></tbody></table> <h3>General</h3> <p>For every function f : X → Y {\displaystyle f:X\to Y} and all subsets A ⊆ X {\displaystyle A\subseteq X} and B ⊆ Y , {\displaystyle B\subseteq Y,} the following properties hold: </p> <table><tbody><tr><th>Image</th><th>Preimage</th></tr><tr><td> f ( X ) ⊆ Y {\displaystyle f(X)\subseteq Y} </td><td> f − 1 ( Y ) = X {\displaystyle f^{-1}(Y)=X} </td></tr><tr><td> f ( f − 1 ( Y ) ) = f ( X ) {\displaystyle f\left(f^{-1}(Y)\right)=f(X)} </td><td> f − 1 ( f ( X ) ) = X {\displaystyle f^{-1}(f(X))=X} </td></tr><tr><td> f ( f − 1 ( B ) ) ⊆ B {\displaystyle f\left(f^{-1}(B)\right)\subseteq B} (equal if B ⊆ f ( X ) ; {\displaystyle B\subseteq f(X);} for instance, if f {\displaystyle f} is surjective)<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></td><td> f − 1 ( f ( A ) ) ⊇ A {\displaystyle f^{-1}(f(A))\supseteq A} (equal if f {\displaystyle f} is injective)<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></td></tr><tr><td> f ( f − 1 ( B ) ) = B ∩ f ( X ) {\displaystyle f(f^{-1}(B))=B\cap f(X)} </td><td> ( f | A ) − 1 ( B ) = A ∩ f − 1 ( B ) {\displaystyle \left(f\vert _{A}\right)^{-1}(B)=A\cap f^{-1}(B)} </td></tr><tr><td> f ( f − 1 ( f ( A ) ) ) = f ( A ) {\displaystyle f\left(f^{-1}(f(A))\right)=f(A)} </td><td> f − 1 ( f ( f − 1 ( B ) ) ) = f − 1 ( B ) {\displaystyle f^{-1}\left(f\left(f^{-1}(B)\right)\right)=f^{-1}(B)} </td></tr><tr><td> f ( A ) = ∅ if and only if A = ∅ {\displaystyle f(A)=\varnothing \,{\text{ if and only if }}\,A=\varnothing } </td><td> f − 1 ( B ) = ∅ if and only if B ⊆ Y ∖ f ( X ) {\displaystyle f^{-1}(B)=\varnothing \,{\text{ if and only if }}\,B\subseteq Y\setminus f(X)} </td></tr><tr><td> f ( A ) ⊇ B if and only if there exists C ⊆ A such that f ( C ) = B {\displaystyle f(A)\supseteq B\,{\text{ if and only if }}{\text{ there exists }}C\subseteq A{\text{ such that }}f(C)=B} </td><td> f − 1 ( B ) ⊇ A if and only if f ( A ) ⊆ B {\displaystyle f^{-1}(B)\supseteq A\,{\text{ if and only if }}\,f(A)\subseteq B} </td></tr><tr><td> f ( A ) ⊇ f ( X ∖ A ) if and only if f ( A ) = f ( X ) {\displaystyle f(A)\supseteq f(X\setminus A)\,{\text{ if and only if }}\,f(A)=f(X)} </td><td> f − 1 ( B ) ⊇ f − 1 ( Y ∖ B ) if and only if f − 1 ( B ) = X {\displaystyle f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\,{\text{ if and only if }}\,f^{-1}(B)=X} </td></tr><tr><td> f ( X ∖ A ) ⊇ f ( X ) ∖ f ( A ) {\displaystyle f(X\setminus A)\supseteq f(X)\setminus f(A)} </td><td> f − 1 ( Y ∖ B ) = X ∖ f − 1 ( B ) {\displaystyle f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)} <a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></td></tr><tr><td> f ( A ∪ f − 1 ( B ) ) ⊆ f ( A ) ∪ B {\displaystyle f\left(A\cup f^{-1}(B)\right)\subseteq f(A)\cup B} <a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></td><td> f − 1 ( f ( A ) ∪ B ) ⊇ A ∪ f − 1 ( B ) {\displaystyle f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)} <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></td></tr><tr><td> f ( A ∩ f − 1 ( B ) ) = f ( A ) ∩ B {\displaystyle f\left(A\cap f^{-1}(B)\right)=f(A)\cap B} <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></td><td> f − 1 ( f ( A ) ∩ B ) ⊇ A ∩ f − 1 ( B ) {\displaystyle f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)} <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></td></tr></tbody></table> <p>Also: </p> <ul><li> f ( A ) ∩ B = ∅ if and only if A ∩ f − 1 ( B ) = ∅ {\displaystyle f(A)\cap B=\varnothing \,{\text{ if and only if }}\,A\cap f^{-1}(B)=\varnothing } </li></ul> <h3>Multiple functions</h3> <p>For functions f : X → Y {\displaystyle f:X\to Y} and g : Y → Z {\displaystyle g:Y\to Z} with subsets A ⊆ X {\displaystyle A\subseteq X} and C ⊆ Z , {\displaystyle C\subseteq Z,} the following properties hold: </p> <ul><li> ( g ∘ f ) ( A ) = g ( f ( A ) ) {\displaystyle (g\circ f)(A)=g(f(A))} </li> <li> ( g ∘ f ) − 1 ( C ) = f − 1 ( g − 1 ( C ) ) {\displaystyle (g\circ f)^{-1}(C)=f^{-1}(g^{-1}(C))} </li></ul> <h3>Multiple subsets of domain or codomain</h3> <p>For function f : X → Y {\displaystyle f:X\to Y} and subsets A , B ⊆ X {\displaystyle A,B\subseteq X} and S , T ⊆ Y , {\displaystyle S,T\subseteq Y,} the following properties hold: </p> <table><tbody><tr><th>Image</th><th>Preimage</th></tr><tr><td> A ⊆ B implies f ( A ) ⊆ f ( B ) {\displaystyle A\subseteq B\,{\text{ implies }}\,f(A)\subseteq f(B)} </td><td> S ⊆ T implies f − 1 ( S ) ⊆ f − 1 ( T ) {\displaystyle S\subseteq T\,{\text{ implies }}\,f^{-1}(S)\subseteq f^{-1}(T)} </td></tr><tr><td> f ( A ∪ B ) = f ( A ) ∪ f ( B ) {\displaystyle f(A\cup B)=f(A)\cup f(B)} <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></td><td> f − 1 ( S ∪ T ) = f − 1 ( S ) ∪ f − 1 ( T ) {\displaystyle f^{-1}(S\cup T)=f^{-1}(S)\cup f^{-1}(T)} </td></tr><tr><td> f ( A ∩ B ) ⊆ f ( A ) ∩ f ( B ) {\displaystyle f(A\cap B)\subseteq f(A)\cap f(B)} <a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a>(equal if f {\displaystyle f} is injective<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a>)</td><td> f − 1 ( S ∩ T ) = f − 1 ( S ) ∩ f − 1 ( T ) {\displaystyle f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)} </td></tr><tr><td> f ( A ∖ B ) ⊇ f ( A ) ∖ f ( B ) {\displaystyle f(A\setminus B)\supseteq f(A)\setminus f(B)} <a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a>(equal if f {\displaystyle f} is injective<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a>)</td><td> f − 1 ( S ∖ T ) = f − 1 ( S ) ∖ f − 1 ( T ) {\displaystyle f^{-1}(S\setminus T)=f^{-1}(S)\setminus f^{-1}(T)} <a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></td></tr><tr><td> f ( A △ B ) ⊇ f ( A ) △ f ( B ) {\displaystyle f\left(A\triangle B\right)\supseteq f(A)\triangle f(B)} (equal if f {\displaystyle f} is injective)</td><td> f − 1 ( S △ T ) = f − 1 ( S ) △ f − 1 ( T ) {\displaystyle f^{-1}\left(S\triangle T\right)=f^{-1}(S)\triangle f^{-1}(T)} </td></tr></tbody></table> <p>The results relating images and preimages to the (<a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean</a>) algebra of <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> and <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> work for any collection of subsets, not just for pairs of subsets: </p> <ul><li> f ( ⋃ s ∈ S A s ) = ⋃ s ∈ S f ( A s ) {\displaystyle f\left(\bigcup _{s\in S}A_{s}\right)=\bigcup _{s\in S}f\left(A_{s}\right)} </li> <li> f ( ⋂ s ∈ S A s ) ⊆ ⋂ s ∈ S f ( A s ) {\displaystyle f\left(\bigcap _{s\in S}A_{s}\right)\subseteq \bigcap _{s\in S}f\left(A_{s}\right)} </li> <li> f − 1 ( ⋃ s ∈ S B s ) = ⋃ s ∈ S f − 1 ( B s ) {\displaystyle f^{-1}\left(\bigcup _{s\in S}B_{s}\right)=\bigcup _{s\in S}f^{-1}\left(B_{s}\right)} </li> <li> f − 1 ( ⋂ s ∈ S B s ) = ⋂ s ∈ S f − 1 ( B s ) {\displaystyle f^{-1}\left(\bigcap _{s\in S}B_{s}\right)=\bigcap _{s\in S}f^{-1}\left(B_{s}\right)} </li></ul> <p>(Here, S {\displaystyle S} can be infinite, even <a href="/facts/Uncountably_infinite/zl2WqyJQ">uncountably infinite</a>.) </p><p>With respect to the algebra of subsets described above, the inverse image function is a <a href="/facts/Lattice_homomorphism/5ak6ufky">lattice homomorphism</a>, while the image function is only a <a href="/facts/Semilattice/kW5po0rp">semilattice</a> homomorphism (that is, it does not always preserve intersections). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bijection%2c_injection_and_surjection/ayC7YE3z">Bijection, injection and surjection</a> – Properties of mathematical functions</li> <li><a href="/facts/Fiber_(mathematics)/rL3pwFwa">Fiber (mathematics)</a> – Set of all points in a function's domain that all map to some single given point</li> <li><a href="/facts/Image_(category_theory)/r82bXCQc">Image (category theory)</a> – term in category theoryPages displaying wikidata descriptions as a fallback</li> <li><a href="/facts/Kernel_of_a_function/co7bK7Sk">Kernel of a function</a> – Equivalence relation expressing that two elements have the same image under a functionPages displaying short descriptions of redirect targets</li> <li><a href="/facts/Set_inversion/C4UNxWIx">Set inversion</a> – Mathematical problem of finding the set mapped by a specified function to a certain range</li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (1991). <i>Algebra</i>. Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 81-203-0871-9.</li> <li>Blyth, T.S. (2005). <i>Lattices and Ordered Algebraic Structures</i>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-85233-905-5..</li> <li>Dolecki, Szymon; Mynard, Frédéric (2016). <i>Convergence Foundations Of Topology</i>. New Jersey: World Scientific Publishing Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4571-52-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/945169917">945169917</a>.</li> <li><a href="/facts/Paul_Halmos/mR11lcap">Halmos, Paul R.</a> (1960). <a href="https://archive.org/details/naivesettheory0000halm"><i>Naive set theory</i></a>. The University Series in Undergraduate Mathematics. van Nostrand Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780442030643. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0087.04403">0087.04403</a>. {{cite book}}: ISBN / Date incompatibility (help)</li> <li>Kelley, John L. (1985). <i>General Topology</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 27 (2 ed.). Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90125-1.</li> <li><a href="/facts/James_Munkres/Ybfa6fs4">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). <a href="/facts/Upper_Saddle_River%2c_NJ/Ps08aEYT">Upper Saddle River, NJ</a>: <a href="/facts/Prentice_Hall%2c_Inc/bynWAuat">Prentice Hall, Inc</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-181629-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/42683260">42683260</a>.</li></ul> <p><i>This article incorporates material from Fibre on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 2019-11-05. Retrieved 2020-08-28. <a href="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" target="_blank">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</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Sect.8 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 2020-08-28. <a href="https://mathworld.wolfram.com/Image.html" target="_blank">https://mathworld.wolfram.com/Image.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Dolecki & Mynard 2016, pp. 4–5. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. <a href="https://search.worldcat.org/oclc/945169917" target="_blank">https://search.worldcat.org/oclc/945169917</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Blyth 2005, p. 5. - Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S. <a href="/wiki/Jean_E._Rubin" target="_blank">/wiki/Jean_E._Rubin</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2 <a href="https://web.archive.org/web/20180207010648/https://pdfs.semanticscholar.org/d8d8/5cdd3eb2fd9406d13b5c04d55708068031ef.pdf" target="_blank">https://web.archive.org/web/20180207010648/https://pdfs.semanticscholar.org/d8d8/5cdd3eb2fd9406d13b5c04d55708068031ef.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hoffman, Kenneth (1971). Linear Algebra (2nd ed.). Prentice-Hall. p. 388. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>See Halmos 1960, p. 31 - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>See Munkres 2000, p. 19 - Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. <a href="https://search.worldcat.org/oclc/42683260" target="_blank">https://search.worldcat.org/oclc/42683260</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>See Halmos 1960, p. 31 - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>See Munkres 2000, p. 19 - Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. <a href="https://search.worldcat.org/oclc/42683260" target="_blank">https://search.worldcat.org/oclc/42683260</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>See Halmos 1960, p. 31 - Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. <a href="https://archive.org/details/naivesettheory0000halm" target="_blank">https://archive.org/details/naivesettheory0000halm</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Kelley 1985, p. 85 - Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. Vol. 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1. <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Kelley 1985, p. 85 - Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. Vol. 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>See Munkres 2000, p. 21 - Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. <a href="https://search.worldcat.org/oclc/42683260" target="_blank">https://search.worldcat.org/oclc/42683260</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>See Munkres 2000, p. 21 - Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. <a href="https://search.worldcat.org/oclc/42683260" target="_blank">https://search.worldcat.org/oclc/42683260</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> </ol>