Editing Image (mathematics) (section)

=== 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).