Talk:Main Page

Dirichlet Series
So I'm working on expanding the Dirichlet series category and adding all the important theorems and such; however, much of theory can be proven in the broader sense of general Dirichlet series so I believe the current Categories are unsuitable. I'm planning on doing some restructuring with General Dirichlet Series as a Category and Ordinary Dirichlet Series as a subcategory, with the general theorems having a 'subpage' for the form the theorem takes in the ordinary sense. Let me know if I mess anything up, or if the wiki would prefer a different method of doing this. --AliceInNumberland (talk) 22:23, 26 May 2018 (EDT)


 * Categories are good, I think.


 * One point to bear in mind: try to provide a source for your work, particularly if you are adding definitions. Look at the "Sources" section at the bottom of most pages to see what I mean. Yes, I know that a lot of the pages in this area do not have such sources, but that's because some contributors have in the past preferred not to follow our house rules. Work will of course need to be done to rectify this. --prime mover (talk) 01:13, 27 May 2018 (EDT)


 * Also, please don't just replace stuff -- if you are expanding on an existing proof, please try and leave the existing shape up as far as possible, including retaining any existing lemmata. If what you are doing is completely new, then add a second proof, and put a comment in the talk page if you need to discuss the viability of the existing one. This is specifically in reference to Dirichlet Series Convergence Lemma, in which reference to the sub-lemma has been lost.


 * You will notice I have moved your new theorem proof to a new page discussing the general case, subpaged from the original. --prime mover (talk) 03:07, 27 May 2018 (EDT)

A new shortcut
We have had the $\LaTeX$ shortcut \rd defined for some time which is basically \, \mathrm d. This is convenient to use as the "d" in integrals. But for derivatives it is not so useful, because the extra gap at the start extends the fraction line and makes it look substandard.

Hence I have added \d, which is just \mathrm d. Thus we have, for example, \dfrac {\d y} {\d x} which produces $\dfrac {\d y} {\d x}$.

I have gone through replacing all instances that I can find of \dfrac {\rd y} {\rd x} (and all similar constructs) with their equivalent. --prime mover (talk) 16:30, 2 August 2018 (EDT)

Volume of paraboloid
After a few long years of search and failure, I was finally able to prove that the volume of a paraboloid is half of its circumscribed cylinder through Cavalieri's Principle.

It is in an analogy to the hemisphere proof, but somewhat harder because you need to use the XY-plane and find the right parabola functions.

In the sketch below, a circle with the green radius shares equal areas with the blue line ring for equal height cross sections.

https://www.desmos.com/calculator/j52jk2gdnt

Can someone please create this page and a 3D-sketch proof? Simcha Waldman (talk) 06:15, 5 August 2018 (EDT)


 * You are invited to create the page yourself. --prime mover (talk) 06:32, 5 August 2018 (EDT)

New $\LaTeX$ macros for your convenience and our internal consistency
There are a number of new $\LaTeX$ macros which have been developed recently as a result of a lot of discussion some time back which never ended up happening at the time.

They can be found on the page Symbols:LaTeX Commands/ProofWiki Specific, transcluded here:

Others may happen as and when I think of them. --prime mover (talk) 03:48, 25 August 2018 (EDT)


 * This is awesome! Thanks!! — Timwi (talk) 19:27, 27 August 2018 (EDT)

Refactoring of Combination Rules for Sequences
As you may have noticed, work is in progress refactoring the Combination Theorem for Sequences section.

This is ongoing, and in the meantime the pages will look a mess and links will be broken.

Please bear with me, this is being fitted in around a change to my work schedule in which I have been told I am now contracted to work an extra hour a day, so I have one less hour in the mornings to do this stuff. Apologies. --prime mover (talk) 02:24, 4 September 2018 (EDT)

Breaking news
https://aperiodical.com/2018/09/michael-atiyah-claims-proof-of-riemann-hypothesis/


 * Riemann hypothesis likely remains unsolved despite claimed proof -- Timwi (talk) 12:10, 28 September 2018 (EDT)

Proofs Wanted category
So... I realize it is a tad late for this suggestion, but would it perhaps be a good idea to have separate proofs wanted (sub)categories for analysis, algebra, probability theory, discrete maths, chess, ...? The category currently contains 1700 pages and this is sure to grow even more over time. Even if we don't go back and categorize all these 1700 pages more carefully, it could help if we did this for future pages, so that new collaborators can more easily find proofs that are wanted in their area of interest. KarlFrei (talk) 04:40, 27 September 2018 (EDT)


 * Unfortunately the "Proofs Wanted" category is added automatically by the ProofWanted template, and to subcategorise these would require some work. I'm not saying it won't happen, but it would have to wait till someone has enough headspace.


 * In the meantime feel free to browse around the "Maintenance Needed" category to see if there is anything there you might like to get involved in. There are several categories, some of which are more intellectually accessible than others. --prime mover (talk) 06:02, 27 September 2018 (EDT)

Bots?
It occurs to me that it would be useful to have bots. For instance, a bot to replace a matching pair of left and right parentheses by the paren command. Has there been any work done in this direction? KarlFrei (talk) 11:28, 4 October 2018 (EDT)


 * Not so far.


 * It would be useful, but then contributors have not always adhered to a standard code style. Some have insisted on using their own preferred style, and some have even pathologically refused to write in any consistent sort of style at all. So programming it to find the edge cases may be challenging. --prime mover (talk) 11:34, 4 October 2018 (EDT)


 * I would first have to figure out how to program bots at all... Still, this could be fun. KarlFrei (talk) 06:53, 5 October 2018 (EDT)


 * I have been looking into this, even managed to get beyond the initial setup (key word: pywikibot). I do agree with PM that it will be hard to identify the use cases. More so since parentheses are among the most overloaded symbols in mathematics (denoting sequences, ideals, modulo classes, equation numbering, ...).
 * Since we are moving in a direction of semantics, it will unfortunately come down to tough hand work. What we can possibly do, is establish certain wishes that help us to clean up the site.
 * Subsequently I, or other people, might be able to program out the relevant bots and make them work. Running them is another matter entirely of course.
 * In this direction I am concretely thinking about e.g. an overview of headings used (and on which pages), subpage structure, categories which do not conform to established template usage... the list goes on. The benefit of such lists is that the human mind is good at doing the same thing repeatedly, more so than doing different things each second. So even when we cannot automate the resolution, it might be possible to streamline the efforts. I would be interested in taking this discussion further. As a start, let's make a list of what could possibly be beneficial (so disregarding any feasibility concerns; the proof writer bot is welcomed ;) ). &mdash; Lord_Farin (talk) 05:35, 7 October 2018 (EDT)


 * Further to LF's comment: "we are moving in a direction of semantics" -- some progress is being made with the various usages of $\paren \cdot$, for example:
 * for "structure", e.g. Algebraic Structure, Topology etc.
 * for "ordered pair, ordered tuple"
 * and of course  for general parenthesised entries.
 * But this does not go as far yet as I would like. Still in mind is  for $f \paren x$, and I would love to think up a sufficiently compact key for replacing , so any suggestions there are welcome.


 * Similar for the various usages of $\left\langle {\cdot}\right\rangle$. But the various usages of $\sqbrk \cdot$ are not so easy to name and identify.


 * Advice and suggestions all welcome. --prime mover (talk) 06:05, 7 October 2018 (EDT)

I would like to err on the side of caution in these matters. There is a danger of PW becoming idiosyncratic, only speaking its own dialect. I don't think we are there at this moment, but things like  would bring the danger closer.

Concretely I would like to suggest that we stick to the low-hanging fruit (equivalence classes and interval notation come to mind, as do parentheses given our house style convention). My guess is that there is plenty of low-hanging fruit. Bots could potentially help us in applying some of the improvements, but this has to be investigated on an individual basis. &mdash; Lord_Farin (talk) 14:09, 7 October 2018 (EDT)


 * Okay I take your point ... is idiosyncratic inherently bad, though? I have started experimenting with  for $\map f x$ which I find does make reading a page's $\LaTeX$ easier to follow. But I can reverse these out if you think it's too far.


 * Other low hanging fruit: the various constructs with  are the main ones I'm looking at. --prime mover (talk) 17:46, 7 October 2018 (EDT)


 * Address  calls sounds great. To your important point: I agree that idiosyncrasy has its place and can be outweighed by other benefits (such as legibility and maintainability), see also our heavy usage of templates. However in the case of \map I'm not so convinced somehow. At least not at this moment. Maybe because the construct $f(x)$ is so prevalent also when $f$ is not really a map (functional, measure, propositional function, operator, you name it) and this can easily lead to the construct being used in non-semantic ways (also strictly speaking $f(x)$ is the "application of the mapping $f$ to the element $x$", which "map" can be argued to not convey sufficiently). But as I said, I can totally imagine that over time my position will change. &mdash; Lord_Farin (talk) 12:57, 8 October 2018 (EDT)

Users
I know, this is not particularly urgent or important... I guess I like cleaning up.

The user list at Special:ListUsers contains 3456 users (!), but the vast majority of these are accounts with zero activity. I realize that there is a separate Active users list, but still, maybe accounts with zero contributions that are more than, say, two years old could just be deleted? This is just junk which is cluttering the list.

Alternatively, it would perhaps be possible to modify the software so that only users with at least one edit are shown on the list of users. KarlFrei (talk) 07:00, 5 October 2018 (EDT)


 * If my account were suddenly deleted for no reason other than having made no edits in 2 years, I’d be furious and I’d probably never contribute again. I’m worried that you’re even entertaining the thought. What you should entertain instead if you find the list unhelpful, is automatically filtering the list, not to delete accounts of users who aren’t doing anything wrong. — Timwi (talk) 15:39, 5 October 2018 (EDT)


 * I have never advocated deleting accounts (except, as I say, for those of some obvious spam-attacks we had some time ago). Your contributions have been valued.


 * But there is an outstanding question: why would anyone want to spend time scanning through the list of user accounts in the first place? What does a person get out of it?


 * (Why did you delete my post, BTW?) --prime mover (talk) 16:10, 5 October 2018 (EDT)


 * That was not intentional, I do apologize. It must have been a conflict from editing the page at the same time. I only meant to post the above message, which was in response to KarlFrei. I will restore the messages I accidentally deleted below. — Timwi (talk) 21:06, 5 October 2018 (EDT)


 * No worries. These things do happen, I gather -- but never understood how. The timestamps don't eve match! --prime mover (talk) 04:12, 6 October 2018 (EDT)


 * I am sorry, I did not mean to upset anybody. I think that there has perhaps been a misunderstanding. I never suggested to delete accounts that were temporarily inactive. I only suggested to delete accounts that never contributed anything at all AND were more than two years old. It was only an idea for a minor cleanup, which I thought should be relatively uncontroversial. As I wrote right at the start, it is not particularly important or urgent. KarlFrei (talk) 08:45, 6 October 2018 (EDT)


 * Never mind, I just spotted the checkbox. Still, my view is that these accounts should just be deleted. KarlFrei (talk) 07:07, 5 October 2018 (EDT)


 * There are a number of contributors who contribute but rarely. More than once has a contributor who has not been seen for years sprung back to life and started contributing once more. We have no intention of deleting any accounts (except perhaps we could weed out some of the blocked accounts of the spammers who plagued us for a while). As far as I know, the response of the website is not adversely affected by the fact of there being a large number of accounts. --prime mover (talk) 07:15, 5 October 2018 (EDT)

Speaking of cleaning up, it is probably time to archive this page again. KarlFrei (talk) 07:12, 5 October 2018 (EDT)

Rules
(continuing the discussion from my user page)

Thank you very much for your responses. Allow me to make a few points.


 * 1) I am all in favour of having different proofs for the same result on this website. Indeed, I started off by adding some alternative proofs myself! It is part of the beauty of mathematics that many things can be proven in completely different ways.
 * 2) It would be fantastic if we could indeed have a reasonable discussion about how much alteration of proofs can be allowed. I was honestly getting the impression that the allowed amount of alteration is zero (apart from bringing it into house style etc.) and furthermore that this is not open to discussion.
 * 3) As to why I want to change some proofs: I want to make them more beautiful. If I can only achieve this by having the nice proof side by side with the ugly version, then this seems rather pointless. Moreover, it potentially leads to a situation where we end up having 10 or 20 different proofs of the same result that each only differ in one or two steps.
 * 4) From now on, I will add maintenance tags to any new pages that I create, and hope that that will be sufficient. I will stick to the house style (rules) that I can remember but I am not going to memorize all of them.
 * 5) Even though I formulated my list of house rules in a rather sarcastic manner, I do believe it would be a giant improvement if we pointed these out (written in a more neutral tone of course) to new editors. Lord_Farin himself wrote that this was a familiar pattern for new versus old users. If the new users could at least be pointed to prior discussions on the same topic, or just plain informed about the secret rules BEFORE they break them, I believe this could save a lot of frustration as well as hopefully avoid repetitive discussions.
 * 6) At the very least, the help pages should be expanded. In my view, there should be nothing about this website that an editor could not find out by reading the appropriate help pages. For instance a brief section about Notes and why they are discouraged. The policy makes sense, so why hide it? There are by now 10 years of discussions on these pages and contributors cannot be expected to read all of that.
 * 7) Shouldn't authorship of pages be irrelevant on a wiki?

Let me conclude by saying that I think this is a fantastic website and I believe it can really help people understand proofs, as every step can ideally be clicked on and further explained. This is a structure which cannot be achieved in books at all (well, you would have to do a LOT of page turning...) and one which I expect students would love. (I cannot test this hypothesis myself as I am not in an English speaking country.) I am also very impressed by the huge amount of work that has been put into this site already.


 * You're in the minority. The general consensus of opinion is that this site is appallingly bad, and should not exist. --prime mover (talk) 03:21, 12 October 2018 (EDT)


 * You've told me this before :-) Recall however that you could not find any sources for this claim the last time! I am not so sure that I am in the minority. KarlFrei (talk) 03:29, 12 October 2018 (EDT)


 * Minority of one. --prime mover (talk) 03:55, 12 October 2018 (EDT)

For me, the main goal of contributing here is for the benefit of humans, as well as satisfying my desire for beauty. (However, the website appears to also have some secret goals as well as secret rules, but that is a topic for another day.) KarlFrei (talk) 03:07, 12 October 2018 (EDT)


 * What we are experiencing here is frustration expressed by a new contributor who is disappointed that things aren't what he wants them to be. --prime mover (talk) 04:51, 13 October 2018 (EDT)


 * Ad 2: I think this would be good. I wonder if there is anything that can be meaningfully said to it, though... Maybe a discussion on individual basis is always required, and more efficient than trying to come up with some formulation that will not entirely capture "the policy".


 * A proof as presented is often as it appears (or isomorphic to how it appears) in a particular source work. A major restructuring of such a proof is undesirable because then what is on the page no longer matches the source work cited on that page. Minor tinkering with the presentation is fine; again we have a house style which we prefer to stick to, for example: "$A$ therefore $B$ (because of result $C$)" is suboptimal, we prefer the form "We have $A$. "From result $C$ it follows that $B$". Once we have arranged the proof into that form, there is little "improvement" that can be done which does not constitute a rewrite. And a rewrite, I contend, is to be added as a separate page.


 * Bottom line is: unless you are bringing the style up to scratch, you should not need to be amending proofs that are already there. --prime mover (talk) 04:51, 13 October 2018 (EDT)


 * Yes, you have mentioned this before, and this (the last line in particular) is what I have now written into the help pages. The first paragraph is another point that could/should definitely be mentioned [|there]. Please do this as I can no longer edit that page. I think it would help avoid confusion in the future.
 * I am still trying to understand the reasoning behind this policy. (It is important for me to understand things.) Why do we stick so closely to source works? Is it just because you like it that way? I have seen you do this with a proof from a book that you really hated at one point, and I do not understand this seemingly undue deference to apparently low quality sources. KarlFrei (talk) 05:30, 13 October 2018 (EDT)


 * Ad 4: This will indeed suffice :).


 * You should not need to add maintenance tags. This is usually done by someone coming after you who is in a position to inspect the code and add the tags as necessary. Yes, there are people who (soon after starting to contribute here) have been producing pages which adhere to house style practically right from the start. If you feel unable to do this, or are unwilling to do this, then yes, feel free to add whatever tags you consider may be necessary -- although if you have already identified non-compliances on the pages you generate, why not address these matters yourself? --prime mover (talk) 04:51, 13 October 2018 (EDT)


 * Ad 5/6: It would be great if you could suggest specific improvements/expansions (as you have already started to do). Maybe entire new pages need to be written.


 * I am glad that you liked my changes to the help pages. KarlFrei (talk) 05:30, 13 October 2018 (EDT)


 * On the other hand, the pages are already large enough to be unwieldy. It would be hoped that by becoming familiar with the site (by using it, and by finding out what is already there), one can get a good idea of the general intent of what we are doing here. If a contributor has a completely different idea of what he or she thinks this site should be, that is a completely different discussion. But (purely as a point of good manners) one should at least make an attempt to work within the given framework.


 * Maybe we can relegate other material to different pages and focus on the main points. Maybe we should limit the first paragraph to mentioning exactly this ("Just walk along, see what we do. For specific guidelines, see below."). Maybe we can identify some articles that express this in a good way. Seeing the pattern of misunderstanding, we might consider a change to make the expectation more explicit. (NB. Obviously I am not advocating that anyone can just do whatever they want, but instead that we could experiment to find out if we can do a better way of communicating what the way of working is.) &mdash; Lord_Farin (talk) 06:27, 13 October 2018 (EDT)


 * That sounds like a good idea. Perhaps we should have started with this. Never anticipated we'd ever have trouble of this sort of magnitude. --prime mover (talk) 11:10, 13 October 2018 (EDT)


 * Ad 7: It is my opinion that structural, comprehensible exposition of mathematical content requires some deeper ideas and in some sense, a plan. If the author of the "plan" is around, why not ask? If the content is already there, why not outline and discuss the new plan somewhere? (Of course this only applies to "material changes"; but see 2. above). &mdash; Lord_Farin (talk) 04:05, 13 October 2018 (EDT)

16000
Congratulations to all on reaching a nice round number of proofs again! Now, on to the next proof... KarlFrei (talk) 06:05, 13 October 2018 (EDT)


 * Most of those are probably redirects or aggregation pages, or notes pages. Very few of them are actually proofs as such. --prime mover (talk) 18:40, 13 October 2018 (EDT)

Suggested change in approach to section spacing
All, I've been following up on an old idea to simplify section spacing. Concretely, to eliminate the need to have multiple whitelines in a page's source to achieve the desired spacing.

Concretely, it involves the following steps:


 * 1) As seen on User:Lord_Farin/common.css, the section spacing for H2 headings (except the first one) has been increased to 2.5em, which is 1.5em</tt> more than it presently is. I arrived at this number by a visual experiment. I've tried various levels of zoom and the effect is consistently the same as having an extra white line in the source code.
 * 2) To correct the existing white lines (avoiding double whitespace), there is a tiny bit of JavaScript User:Lord_Farin/common.js. It removes empty paragraphs before H2 headings (i.e. those created by double white lines in the source code).
 * 3) To fix the issue structurally (so that we do not need the JS to do work every time a page is loaded), I have created a script for the experimental bot that I have been configuring. The effect has been demonstrated in the following edit: User:Lord_Farin/Bot.

To demonstrate the effect of step 1. compared to the existing practice, you can use User:Lord_Farin/Sandbox, and the "inspect element" functionality of your browser (F12 usually) to adjust the spacing of the left heading to 2.5em</tt>.

For me, the effect of 1. + 2. is that both samples look completely identical.

With the aid of the bot described in 3., we can start applying 1. to all existing pages in an automated fashion (additionally giving a proof of concept fur further automation).

What are your thoughts? &mdash; Lord_Farin (talk) 10:03, 14 October 2018 (EDT)

NB. To minimise impact, the current changes to CSS and JS only apply to user space. I will take care of adjusting this before moving them over to their MediaWiki counterparts. NB2. I have checked and in last months' reports I could not find browser sessions which didn't support the technology involved in this adjustment, so from that perspective we are also fine.


 * Quick thoughts ...
 * a) Why not H3 headings too?
 * b) We have a style that evolved wherein the equivalence proofs resulted in multiple definitions not being presented with the same gap as on the Definition page itself.
 * No other things come to mind. --prime mover (talk) 16:17, 14 October 2018 (EDT)


 * Two birds, one stone: Because the definitions in b) are transcluded using H3 headings. &mdash; Lord_Farin (talk) 16:25, 14 October 2018 (EDT)


 * This plan sounds great! Go for it, I would say. (Perhaps let the bot run for only 10 seconds or so first, to see what it does.) Any kind of automation that we can build in is great. Next on my personal wishlist would be automated linking of definitions (only for uniquely defined concepts of course). KarlFrei (talk) 09:25, 22 October 2018 (EDT)


 * Incidentally, Wikipedia has a bot which automatically signs unsigned posts on talk pages (SignBot?). We may want to start using that as well (if we can just copy it over). KarlFrei (talk) 09:27, 22 October 2018 (EDT)

Theorem of Applied Group Theory
Have you noticed that when you go through a door (with a traditional "rotary" lock on it), unlocking it to go through and then locking it again when you've passed through), you turn the key in the same direction both times?

I mentioned this fact on my facebook page and I had a number of people replying that I am incorrect.

I believe this can be modelled as an application of group theory.

Should this fact also be added to the Veridical Paradoxes category? --prime mover (talk) 07:01, 4 November 2018 (EST)


 * You are definitely right. It hasn't struck me as paradoxical, though. But maybe that's just me. &mdash; Lord_Farin (talk) 14:27, 5 November 2018 (EST)

Help needed with Subset Products of Normal Subgroup with Normal Subgroup of Subgroup
Anyone with a good level of skill in group theory who can finish off (or completely rewrite) the above proof?

I can't work out how to prove that if $H$ is a subgroup of $G$, and $N$ is normal in $G$, and $K$ is normal in $H$, then $NK$ is normal in $NH$. I seem to need to prove that $K$ is normal in $N$ which is not where I wanted to go. Anyone? --prime mover (talk) 08:06, 5 November 2018 (EST)


 * The solution is to note that $K \le G$, and then use $g N = N g$ from $N \lhd G$ to infer $K N = N K$. I've completed the proof using this approach. &mdash; Lord_Farin (talk) 14:45, 5 November 2018 (EST)


 * !?! Looked right past it. Good job. --prime mover (talk) 15:03, 5 November 2018 (EST)

Possible error in Inverse Hyperbolic Sine is Odd Function
Hello! I am new and hope that this is the right place to ask. The Help:Editing page discourages editing existing proofs as a newbie. But as regards Inverse Hyperbolic Sine is Odd Function, I think that the last line of the proof is wrong. It should instead be: $y = - \sinh^{-1} x$. Can I modify it this way?


 * Feel free. I removed the offending section of Help:Editing, which was unsanctioned.


 * You are correct -- there is indeed a silly mistake on the page in question. --prime mover (talk) 13:26, 24 November 2018 (EST)


 * Thank you so much.

== How to avoid plagiarism while obeying Help:Editing/House Style/Sources

The inspiration for a proof I wrote came from an online source not listed in Help:Editing/House Style/Sources. I wrote the proof. I filled in all the details and organized the order of statements. But once I saw the concepts mentioned in the online source, it was all obvious, just a matter of detail work. How do I obey the citation policy without plagiarism? I am told that in academic work one must always reveal the source of ones ideas. Can I reveal where I got the ideas without making a "general citation source"? Thank you --Pelliott (talk) 22:25, 25 January 2019 (EST)

How to avoid plagiarism while obeying Help:Editing/House Style/Sources
I got the inspiration for a proof from a online source not mentioned in Help:Editing/House Style/Sources. I wrote the proof. I filled in all the details and ordered all the statements. But it was all obvious once I saw the concepts mentioned together in my online source, just detail work. I am told that in academic work, one must always reveal all the sources for one's ideas. How can I do this? Can I reveal where I got the idea without making a "general citation source"? Thank You --Pelliott (talk) 22:38, 25 January 2019 (EST)


 * Is this an academic work? I never thought so. It's just an amateur web page. --prime mover (talk) 05:21, 26 January 2019 (EST)

Things that are missing or hard to find.
This is a list of things that are missing or hard to find. Pythagorean Identity it's there somewhere but hard to find. Proof that reals are uniquely embedded in complexes as a normed field. Slope-intercept form of equation of straight line. Two-point form of equation of straight line. I will add to list as I find more. --Pelliott (talk) 16:03, 2 February 2019 (EST)

Account request
Can we remove the word length constraint on the account request biog? I get the feeling that prospective contributors consider 50 words is a bit of an imposition. I would be tempted to agree with them, I'm afraid.

All account requests need moderator approval, and it should be no problem for one such moderator to actually make a judgment call as to whether the request is genuine. And if the occasional robot spammer gets through, well, no lasting harm will be done as long as at least one contributor is borderline vigilant. --prime mover (talk) 03:02, 11 March 2019 (EDT)


 * Sure, I'll change it to be 1. --Joe (talk) 11:19, 11 March 2019 (EDT)


 * I think now we're in the sweet spot. --prime mover (talk) 02:49, 12 March 2019 (EDT)

There is a typo at the page https://proofwiki.org/wiki/Image_of_Union_under_Mapping
The last line of proof 1 states "$𝑓[𝐴]∪𝑓[𝐵] \subseteq 𝑓[𝐴∪𝐵]$ by the Definition of Set Equality". '$\subseteq$' should be '='.

I tried to correct the typo, but didn't know how. Please update the page. Thanks.