Editing Image (mathematics) (section)

== Properties ==

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

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

=== General ===

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

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

Also:

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

=== Multiple functions ===

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

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

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

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

{| class="wikitable"
|-
! Image
! Preimage
|-
|<math>A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)</math>
|<math>S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)</math>
|-
|<math>f(A \cup B) = f(A) \cup f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985">{{harvnb|Kelley|1985|p=[{{Google books|plainurl=y|id=-goleb9Ov3oC|page=85|text=The image of the union of a family of subsets of X is the union of the images, but, in general, the image of the intersection is not the intersection of the images}} 85]}}</ref>
|<math>f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)</math>
|-
|<math>f(A \cap B) \subseteq f(A) \cap f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985" /><br>(equal if <math>f</math> is injective<ref name="munkres-2000-p21">See {{harvnb|Munkres|2000|p=21}}</ref>)
|<math>f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)</math>
|-
|<math>f(A \setminus B) \supseteq f(A) \setminus f(B)</math><ref name="lee-2010-p388" /><br>(equal if <math>f</math> is injective<ref name="munkres-2000-p21" />)
|<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math><ref name="lee-2010-p388" />
|-
|<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math><br>(equal if <math>f</math> is injective)
|<math>f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)</math>
|-
|}

The results relating images and preimages to the ([[Boolean algebra (structure)|Boolean]]) algebra of [[Intersection (set theory)|intersection]] and [[Union (set theory)|union]] work for any collection of subsets, not just for pairs of subsets:

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

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