Editing Image (mathematics) (section)

==<span id="Notation">Notation</span> for image and inverse image==
The traditional notations used in the previous section do not distinguish the original function <math>f : X \to Y</math> from the image-of-sets function <math>f : \mathcal{P}(X) \to \mathcal{P}(Y)</math>; 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{{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>
* <math>f^\leftarrow  : \mathcal{P}(Y) \to \mathcal{P}(X)</math> with <math>f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}</math>

===Star notation===

* <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><ref>{{Cite book |last=Hoffman |first=Kenneth |title=Linear Algebra |publisher=Prentice-Hall |year=1971 |edition=2nd |pages=388 |language=en}}</ref> but this usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>