Talk:Image (mathematics)

Untitled

The epsilon signs don't show up for me (show up as boxes)

You need a better browser (or default font). We should probably include a link to {{SpecialChars}} on this page. -- Fropuff 21:45, 8 December 2005 (UTC)[reply]

Image and graph

Using the set names given at the beginning of this articles, do all elements of the set Y ordered with all elements of the set X make up the graph of the function f? --Abdull 15:46, 31 May 2006 (UTC)[reply]

I'm not sure what you mean by "all elements of the set Y ordered with all elements of the set X". Do you mean X x Y, that is the set of all ordered pairs (x, y) such that x is an element of X, and y is an element of Y? IIn any case, that is not the graph of f. The graph of f is the set of all pairs (x, f(x)) such that x is an element of the domain of f. Thus the graph of f is a subset of X x Y, (in fact it is a subset of X x f[X]). Paul August ☎ 16:08, 31 May 2006 (UTC)[reply]

Image for image

I think that an illustration would be very helpful in describing this concept. Below is an image that I made for this article. I used terminology not used in the article however. Silly Rabbit, could you tell me what variables you want me to replace in the image. Thanks--Cronholm144 05:18, 31 May 2007 (UTC)[reply]

references for consequences

Does anyone know of any references for the identities listed in the "consequences" section? Deriving the first four identities is an exercise in a book I'm reading, and I'm able to correctly demonstrate all but the second. I can only show that it's an equality, but that's apparently wrong. If anyone knows where I can find its derivation, please update the references section, or include the derivation here in the talk section. Thanks. --Paul 16:13, 11 June 2007 (UTC)[reply]

Those identities are well known in universal algebra and category theory. I had them in lecture notes for discrete math in the 1970's and they appeared in some discrete math book that gave my lecture notes credit. I will try to find some reference.SixWingedSeraph (talk) 02:17, 18 April 2009 (UTC)[reply]

Consider the following counterexample for why it is not an equality. Let f=x^2 and let E_1=[0,inf), let E_2=(-inf,0]. The intersection of f(E_1) with f(E_2) is [0, inf). The image of f(E_1 intersect with E_2) is {0}. You should be able to use this example to pinpoint why any attempt at a proof for the equality will fail. —Preceding unsigned comment added by 202.36.179.66 (talk) 01:47, 10 June 2008 (UTC)[reply]

inverse and image

It seems that the inverse image of a function and the image of the inverse of a function is written the same way. Very confusing.

I have rewritten the introduction to mention inverse image as well as image. I changed the sentence "when taking each element of the set, successively, as the function's argument" -- it makes the image sound like it is defined over time! SixWingedSeraph (talk) 15:46, 15 April 2009 (UTC) Revised the section on images extensively. SixWingedSeraph (talk) 20:46, 16 April 2009 (UTC)[reply]

square bracket notation for image function

The entry as written uses square brackets f[A] for the image function in contrast to ordinary function evaluation f(x). Then it says f(A) is common when there is no danger of confusion. Where did this square bracket notation come from? I have never seen it before. I have not changed it. SixWingedSeraph (talk) 20:46, 16 April 2009 (UTC)[reply]

It's been over 10 years, but I entirely agree, the notation is bizarre. Since it seems to have been invented for Wikipedia alone, I don't believe it belongs here, despite its admirable lack of ambiguity. 2600:1700:FA10:75C0:B9DA:B53:3233:7387 (talk) 12:07, 6 May 2021 (UTC)[reply]

It is used e.g. in the Halmos' textbook^[1] (Sect.8). - Jochen Burghardt (talk) 11:02, 12 May 2021 (UTC)[reply]

References

^ Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand.

It's a common notation. Definitely not "invented for Wikipedia". I think it's common in real analysis, though I checked Folland and to my surprise he doesn't use it.

Set theorists tend to use

f"A

, which I can't seem to find any way of getting <math> tags to parse. We should probably mention this notation as well. --Trovatore (talk) 17:13, 12 May 2021 (UTC)[reply]

Examples: The Origin, The Origin

"The fibres f −1({a}) are concentric circles about the origin, the origin, and the empty set, depending on whether a>0, a=0, or a<0, respectively."

Is this intentional? If so, it should be simplified to a>=0 and a<0. LokiClock (talk) 12:31, 20 May 2009 (UTC)[reply]

Fixed, thanks to User:99.150.134.117. ᛭ LokiClock (talk) 02:47, 21 August 2011 (UTC)[reply]

Image (set): Proposed redirect

Image (set) is briefer, which aids navigation and editing. Considering browser links will show a redirect's title & not its endpoint, calling the article by this name is a good way to remind people what an image is without diverting them. ᛭ LokiClock (talk) 09:11, 20 August 2011 (UTC)[reply]

Set is the word in the English dictionary with the most meanings. So no I don't think this is a good idea. Dmcq (talk) 11:43, 20 August 2011 (UTC)[reply]

Where there are named sets of the other kinds, they would more logically be categorized under a different name. There are no current conflicts. With image set, it wouldn't make sense to write image (set) for image set, because "image" is not the name of the set when you're talking about a set of images. ᛭ LokiClock (talk) 12:40, 20 August 2011 (UTC)[reply]

I think it would just lead to problems. How about 'Image of a function' like Domain of a function if you really want to change the name for some reason? I don't understand your original rationale about redirects and your second rational has me wondering too. Article names aren't to remind anybody of anything, their purpose is to help them find things. But if you'd like to change to 'Image of a function' I've no objections. Dmcq (talk) 13:02, 20 August 2011 (UTC)[reply]

I would accept Image of a function. Image (mathematics) is used throughout Wikipedia to distinguish the image in question from a graphical impression. Image (set) could be used within mathematics articles to recall the definition of object whose name is metaphorical. This would be a good policy for articles where the terminology is not taken for granted, because people don't have to stop mid-sentence to look at what an image is again. Image of a function may be better than Image (set) for this purpose, except in most contexts I see it explicit that it is an image of a function, but its relationship to the function is only captured by the fact that it's a set. ⁂ Policy: Both are justified under WP:TITLE for recognizability & precision, & WP:R for more specific forms of names. Image of a function provides nothing over Image (mathematics) in conciseness, but it is consistent with Domain of a function. ᛭ LokiClock (talk) 21:43, 20 August 2011 (UTC)[reply]

Links to articles with a bracket in the title like that normally are piped so the thing in brackets isn't visible. There is no need for explanations to be stuck into titles, besides which being a set isn't a particularly special thing in mathematics. Dmcq (talk) 22:48, 20 August 2011 (UTC)[reply]

This may be worth bringing to a policy discussion. The things in the brackets are visible in browsers out of the inline text. My wishful thinking was that the linking article title carry enough information to reward the behavior I see in Wikipedia users, which is hovering over links to see if the title will disambiguate it. This is an obvious reaction to links like "There is no clear boundary between West and East Old Norse." This use is available for articles with names provided for otherwise: Hoy is given the choice between writing Norse and Norse, taking the latter. Yes, being a set isn't all that special. Image (output) would do the trick, though I didn't think of it before. ᛭ LokiClock (talk) 23:13, 20 August 2011 (UTC)[reply]

We should do what people want first and foremost and put in easily recognised names for the topic. There is no bit of WP:Article titles or WP:Disambiguation about explaining things in the title for somebody who is halfway through reading an article. Any such explanation if needed is up to the person writig the article. If you want some facility using hovering rather than clicking on rticle this is not the place - propose it at one of the village pumps like WP:VPR. Dmcq (talk) 23:24, 20 August 2011 (UTC)[reply]

Yes, thank you. ᛭ LokiClock (talk) 00:16, 21 August 2011 (UTC)[reply]

Wikipedia:Village pump (proposals)#Redirects containing topics when possible ᛭ LokiClock (talk) 02:38, 21 August 2011 (UTC)[reply]

Image set

Should it be noted that the Image is sometimes referred to as the 'Image set', that is how it is referred in on the course I'm doing. Also, 'Image set' could redirect to this page? Jonpatterns (talk) 13:13, 7 April 2014 (UTC)[reply]

Notation for image and inverse image: Other terminology

The article says:

Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f.

This looks wrong in multiple ways. First of all, even middle school text distinguish range from codomain in the US. Is it different elsewhere? Second of all, "the image of f" is the relatively rare phrase that should be avoided, whereas the range of f reliably refers to the image of the domain under f. (See ProofWiki for an example of standard usage for functions/mappings.) This section appears to be promoting a somewhat nonstandard and non-referenced usage of the term range as a preferred usage. Cerberus (talk) 11:15, 4 August 2015 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Image (mathematics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs significant expansion, especially the vital role of images in abstract algebra (via the isomorphism theorems for instance). It's possible to create a fork to image (abstract algebra), but some systematic treatment is essential for the project. (If progress is made, please alter my comment accordingly.) Silly rabbit 15:51, 30 May 2007 (UTC) Needs to be kept separate from Range —Preceding unsigned comment added by 71.55.23.120 (talk) 01:02, 5 October 2007 (UTC)[reply]

Last edited at 01:04, 5 October 2007 (UTC). Substituted at 02:14, 5 May 2016 (UTC)

"⊷" operator?

How is the "⊷" (image of) operator used to represent this concept? Shouldn't it be included in this article? --Lance E Sloan (talk) 12:52, 1 November 2017 (UTC)[reply]

I'm entirely unfamiliar with that symbol. Searching for it on Google confirms that it's understood as "image of", but most of the hits seem to be just for the literal English words "image of", and are not mathematics at all.

So I'm guessing it isn't used much. Still, if you can find good sources, and also show that it's genuinely used and not just some ultra-rare idiosyncrasy, then I suppose we can mention it. My cursory search didn't find anything you could use as a source. --Trovatore (talk) 19:45, 1 November 2017 (UTC)[reply]

Terminology

As an undergraduate student who just learned the concept of kernel and image of a certain mapping (especially in the context of group homomorphism), I wondered about something.

Let's say I have a mapping $f\colon X\to Y$ . What's the difference between "range of $f$ " and " ${\text{Im}}\left(f\right)$ " ? Does both of them represent the same thing, but spoken in different way? In other words, are they synonym to each other, and it is redundant to have both of the terminology at the same time ? The Winter Lettuce (talk) 07:08, 20 December 2022 (UTC)[reply]

See § Image of a function and § Other terminology. "Range" may refer, depending on the authors, to either the domain

X

, the codomain

Y

, or to the image

f (X)

. In your context, it is probably used for the codomain. However, a function or homomorphism is surjective if and only if the image equals the codomain. Also, in the context of homomorphisms, the domain and the codomain are often called respectively the source and the target; in this case, "range" refers probably to the image.

Lead

Inspired by Epachamo's recent edits, I suggest the following order for the lead:

Image and preimage of a single element,
image and preimage of a set (subset of the domain and range, respectively),
image and preimage of the function (preimage is rarely, or even never, used here; I'm not sure if we should mention it).

After that, the paragraph about generalization to binary relations should appear. - Jochen Burghardt (talk) 05:34, 12 April 2024 (UTC)[reply]

I'm ok with the new ordering. I emphatically think that we should make mention preimage of a function though, and explicitly point out that it "is rarely, or even never, used here". Epachamo (talk) 16:40, 12 April 2024 (UTC)[reply]

Done - including your mention. - Jochen Burghardt (talk) 15:22, 13 April 2024 (UTC)[reply]

Confusion over image

Image in question and a reddit post discussing it. It seems that this image is either flat out wrong or at least unhelpful and confusing to readers. Obviously, reddit posts aren't reliable sources and all that, but I think this a good enough reason to either modify the caption or image, but I'm not sure how to do either accurately. Pinging @Epachamo, the author of the image.

Based5290 :3 (talk) 05:09, 31 August 2024 (UTC)[reply]

I agree the current caption is misleading. The problem is that it is unclear whether f is shown completely (in which case y' should belong to the blue area), or just an excerpt of f is shown (in which case some other, undepicted, member of X should be mapped to y'). I guess Epachamo intended the latter, but the caption seems to say the former.

Maybe, due to this ambiguity, such abstract Venn diagrams (without referring to a particular example mapping) shouldn't be used here at all. - Jochen Burghardt (talk) 06:48, 31 August 2024 (UTC)[reply]

Thanks for bringing this up. It was at the very least confusing, and the f(x) circle was not necessary. I have changed the image (haha) based on the feedback. Let me know if it makes more sense. Epachamo (talk) 16:39, 31 August 2024 (UTC)[reply]

Looks good to me. Thanks for modifying the image. Based5290 :3 (talk) 16:44, 31 August 2024 (UTC)[reply]

The problem remains: Do you mean X={x,x'} and Y={y,y'}, with no other elements in X and Y? Then the set elements should better be named by some constants (maybe +2,-2 and +4,-4), not by typical variable names. Or do you mean x,x' ∈ X and y,y' ∈ Y, and X and Y contain some non-depicted elements also? Then you need to explain in the caption that all arrows ending at y,y' are shown but no other ones. The same applies to all the Venn diagrams shown in the article. - Jochen Burghardt (talk) 22:30, 31 August 2024 (UTC)[reply]

x,x' ∈ X and y,y' ∈ Y, and X and Y contain some non-depicted elements. I personally think it clear that this is the case, and further explanation in the caption would muddy it even more. Epachamo (talk) 22:50, 31 August 2024 (UTC)[reply]

[1] Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand.

[1]