206
$\begingroup$

How to ask a good question.

This thread has advice on the following aspects of writing a good question on this site. Each item in this list links to an answer below about that specific aspect of question writing.

||||||
$\endgroup$
  • 4
    $\begingroup$ Further, there is this old thread filled with good snippets. $\endgroup$ – Lord_Farin Jun 18 '13 at 9:01
  • $\begingroup$ The physics site developed a similar meta post. $\endgroup$ – DanielSank Jul 28 '17 at 16:00
  • $\begingroup$ I think here we can study the beginning of an edit war in slow-motion (version13 and higher) $\endgroup$ – miracle173 Jan 16 '18 at 18:38
  • 9
    $\begingroup$ @Jack and @Shaun Stop it. Whether ? or . is the correcterer [sorry, had to] choice of punctuation mark is irrelevant. That also applies to the other question. $\endgroup$ – Daniel Fischer♦ Jan 16 '18 at 20:55
  • $\begingroup$ @Shaun: Did you miss Daniel's Fischer comment? It literally consisted of 20% of the comments on this thread (prior to my comment). $\endgroup$ – Asaf Karagila♦ Mar 6 '18 at 23:54
  • $\begingroup$ @AsafKaragila, I only noticed it after I'd made the edit. I'm sorry. $\endgroup$ – Shaun Mar 6 '18 at 23:56
  • 1
    $\begingroup$ @Jack I find the advice to include the source and motivation for the questions quite important. Particularly in the case of contest questions. Users unfamiliar with the contest scene have trouble telling the difference between a contest question and a PSQ, so it is useful to describe the source. Also, any doubts about the contests being current can be cleared at the same time. $\endgroup$ – Jyrki Lahtonen Mar 17 '19 at 20:21
96
$\begingroup$

Provide Context

Context matters. A question can sometimes be answered in one sentence when the discussion is between two experts familiar with each other's background, while the same question may take many paragraphs of detailed computation when being shown to an undergraduate student. By providing a context you help the potential responders to your question give you the best help you need.

Some different ways you can add context to your question

  • Include your work

    You have a question, and if you post it here, you've probably attempted, and failed, to solve it yourself. It is much easier for others to judge the most appropriate "level" for an answer to your question if you provide these attempts. So you'll receive answers better suited to your specific needs.

    Including your work also shows to the community that you're not using this website as an answer machine -- as such, your question will be received more positively.

    A further benefit of writing down precisely what you've tried is that, in the process of doing so, you will very likely spot your error and solve your problem yourself. Bonus!

  • You can provide some motivation to your question.

    Instead of just asking us to find the roots of an equation, tell us where the equation comes from. This is especially the case when your equation comes from models of the physical worlds: those kinds of intuition are great guiding principles for formulating an answer.

  • You can tell us where the question comes from.

    If your question comes from studying a textbook, let us know which book. This way the answers can be phrased in a manner and in a notation more familiar to you. Exposition varies from one book to another, affecting which theorems are appropriate to cite in answers, and which definitions you are starting from (see below).

  • Indicate your own background

    In order to address your question in a useful manner, we need to be able to estimate your background to some degree. (Briefly) Indicate your familiarity with the subject matter so that the answerers have an easier job assessing the audience, and can adjust the level of their answer accordingly.

  • Give full references.

    If you run across a question when reading a scientific paper, be sure to link to that paper using its doi link, or provide a proper bibliographic information. A question that reads "A theorem of Smith says that Widget X is a type of Gadget Y, but I don't see why Property Z must hold" is likely not going to be very comprehensible to other users without telling us which Smith said what when and where.

  • Give definitions.

    Something that you are familiar with may not be so to another user. One should of course use one's best judgment in deciding what objects are sufficiently well-known to not need defining. But when in doubt, either provide the definition or provide a link to a resource that gives the definitions.

    Another case where this can be useful is when the same mathematical object can be defined in many ways, and the answer to your question may depend on the precise definitions used. For example, Widget X may be defined by Author A to satisfy property T. Practically everyone else may prefer to define it as satisfying property S. Showing the equivalence between property T and property S may happen to be one of the harder but lesser known theorems in the past fifty years. If you ask the question, after reading a treatise by Author A, that "Why is property S true for Widget X?"; the common answer "duh, that's by definition" will probably not be very useful to you.

||||||
$\endgroup$
  • 15
    $\begingroup$ I think an answer should demand as little knowledge as possible to understand it regardless the OP's knowledge. This is because each question is not the OP's property but the site's which should serve for the benefits of as many users as possible. Therefore I think giving motivation or context for a question is not necessary as long as it is mathematically clear. $\endgroup$ – user157678 Aug 13 '14 at 0:26
  • 18
    $\begingroup$ @Deep: People capable of writing "mathematically clear" questions are of course free to disregard the suggestions on this page. The fact remains that many questions asked on this site will benefit from (some of) the advice given in this Q+A item. $\endgroup$ – Willie Wong Aug 19 '14 at 7:55
  • 2
    $\begingroup$ As I am just digging into what questions are appropriate and what are not I find the context reason is a fig leaf to close questions that are suspected "homeworks". By definition all pure math questions lack context outside of pure math. This is apparently fine. On the other hand highschool-grade questions are required to provide "motivation" based on real-world applications or the like. If somebody posts a PSQ at highschool level, chances are high it will get downvoted or closed even if it can be clearly answered. If I state a problem at graduate level, this does not happen. Double standards. $\endgroup$ – mol3574710n0fN074710n May 3 '18 at 2:31
  • 5
    $\begingroup$ @mol3574710n0fN074710n: firstly, I emphatically reject your claim that "by definition all pure math questions lack context outside of pure math". Secondly, context does not automatically mean "application". Out of the 6 suggestions given above on how to give context, exactly one has to do with including any "non-mathematical" origins of the question. The remaining five suggestions can be implemented even in a "pure math" setting as you would call it. $\endgroup$ – Willie Wong May 3 '18 at 18:19
  • 1
    $\begingroup$ the problem with the suggestions is that only the first three will apply to the majority of highschool users. I believe when somebody states a clear mathematical problem that can be answered, it should be nobody's buisiness where that problem came from or what motivated the problem. I do believe that because very few higher level math questions are required to provide that information. These "suggestions" are taylored to lower level questions, possibly in an attempt to make sure they are not homework. $\endgroup$ – mol3574710n0fN074710n May 3 '18 at 18:29
  • $\begingroup$ ... to that pure math thing: I should have written: "by definition all pure math questions lack context outside of math". There are of course pure math topics with context in applied math, while it is a defining feature of pure math that it is not applied itself. (The only exception I know of is number theory and encryption) $\endgroup$ – mol3574710n0fN074710n May 3 '18 at 18:32
  • 2
    $\begingroup$ @mol3574710n0fN074710n: ^^ "it should be nobody's business where that problem came from" for pedagogical reasons I disagree. (I find it immensely helpful, and often times necessary, to have my calculus students put their calculus questions in context.) // ^ You ignore many developments in graph theory (computer science), functional analysis (physics, especially solid state and condensed matter), probability theory (finance and operations research), as well as the entire field of partial differential equations. // I find little value in debating this further, so I stop now. $\endgroup$ – Willie Wong May 3 '18 at 19:20
  • $\begingroup$ Ok, just adding, that where I come from neither PDEs nor probability theory would be considered "pure math", all applied - functional analysis would be a border case... this is probably a language problem, maybe that distinction is made differently (or not at all?) in english speaking countries... If that is the case what I have written about "pure math" cannot make any sense indeed... $\endgroup$ – mol3574710n0fN074710n May 3 '18 at 19:27
  • 4
    $\begingroup$ Just adding for future readers, that thanks to a very solid (and patient) introduction the moderator quid has generously given me to the good/poor questions subject, I have been able to clear the misconceptions I had about the site's workings and revised my views stated above... Bottomline: What I commented above was garbage and the expression of some frustration caused by a flawed understanding of the site's workings. In particular there are no "double standards" as I had claimed but sometimes lower quality questions just might get past the "quality checks" due to limited resources (time). $\endgroup$ – mol3574710n0fN074710n May 9 '18 at 1:46
  • $\begingroup$ @WillieWong on etiquette, what's the unwritten rule regarding the asking of a sequence of related questions in your question. Suppose I ask three questions in my post about a proof I made... when or when is it not acceptable etiquette-wise? $\endgroup$ – AmateurMathPirate May 19 '18 at 18:18
  • 2
    $\begingroup$ @AmateurMathPirate: please note that I first summarized these suggestions nearly five years ago by documenting the community norms as it stood then. People have been modifying this (and other) faq documents to reflect the evolving thoughts of the community on this and related matters, and I haven't really been keeping up with the changes myself. So long story short: I don't think I can give you a good answer about this. Maybe you should ask a new question on Meta about it. Either someone will close it as a duplicate and point you to what they consider the current community preference, or you $\endgroup$ – Willie Wong May 21 '18 at 15:42
  • 1
    $\begingroup$ ... get an answer. $\endgroup$ – Willie Wong May 21 '18 at 15:42
  • $\begingroup$ I don't know whether it is a good idea to ask about a proof, a definition, etc. on a book without quoting the context. For example, this question only contains the phrase the questioner doesn't understand, without explaining what $A$ and $\mathfrak p$ are. This is fortunately still understandable. Another worse example contains neither the proposition in question, nor what $A,B,\varphi$ are (rings? a morphism of what?). $\endgroup$ – Yai0Phah Jul 11 '19 at 9:45
54
$\begingroup$

A good title

The title of a question is the first thing people see. Like headings in newspapers, book, song and album titles, their importance is not to be underestimated -- the presence of a good, descriptive title for your question often greatly improves the exposure (and hence the amount and quality of answers) it gets. To ensure maximal descriptiveness of your question's title, review it before posting and ensure that it (still) adequately describes your question's content.

How to choose a good title

  • Make your title your question

    Use your title to convey as much information about your question as possible. Since the tags already convey the general subject area of your question, the title should communicate the question itself as faithfully as possible. If necessary, leave out hypotheses in the title, and in the body of the question, explain why the question requires those hypotheses.

  • MathJax works in titles

    Titles have MathJax support. This means you can e.g. include the integral your question is about in the title, and do not have to resort to vague descriptions like "difficult integral". In your use of MathJax, please adhere to the community guidelines for MathJax in titles. Most importantly, keep the vertical space your title uses to a minimum, and be sure to include at least some plain words.

  • Don't be afraid to make the title long

    Titles are allowed to be anywhere from 15 to 150 characters long. 140 characters (the length of a tweet) of plain text take up about two full lines on the home page, so try to keep it less than that. But 140 characters is a lot longer than you might think. Too many people restrict themselves to 20 character titles. They're trying not to waste your time by making you read a long title, but they end up wasting more of your time because you have to actually open the question to see if it's interesting to you.

  • Make your title interesting for others

    Mathematics.SE is designed to be a repository of good mathematical questions and answers. Thus, there is no need to refer to your personal situation in the title. Make your title a question of universal value. For example, the title Help me solve $a^2+b^2=c^2$ for my exam preparation is very specific to your personal situation. Deriving the formula for Pythagorean triples would be a more universal, better title for the same question.

  • Your question should be clear without the title

    After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title.

Other Tips

When you are posting a question, write your title first. The system will then suggest possible duplicates: take a look at them, opening links in another tab. If none of those are actual duplicates, write out the body of your question. Then go back and put in a better title for the body that you wrote. (From Robert Harvey)

||||||
$\endgroup$
42
$\begingroup$

Avoid "no clue" questions

Too many questions begin or end with "I don't even know how to begin with this problem". While this may be true (you may genuinely have no idea how to approach the problem), it is still not a valid reason to limit your post to the statement of the problem without any mention of your own thoughts. Such questions will most of the time be rejected by the community, which represents a significant waste of time - including for yourself - since the removal of poorly documented questions is not a fully automated process.

There are plenty of ways to get started on a problem when one has "no clue":

  • Write down the definition of the keywords of the problem

This allows you to make sure that you understand them. Use examples ($\star$) when applicable.

Tip: if you realise that you actually don't know the definition of one or several keywords, then you are probably asking the wrong question. However, you now know where to start!

  • If the problem involves formal computations, try with specific settings ($\star$) first

If you are asked to prove that something holds for any value of $n\in \Bbb N$, see if you could prove it when $n=1, 2$ or $3$. Doing that you may observe a repeating pattern, or grounds for a proof by induction.

If the problem involves an arbitrary orthogonal matrix $M$, replace it by your favourite orthogonal matrix. If you realise that you can't write one down, this means that you are not fully at ease with the notions involved, and what's more you know where to start: read about orthogonal matrices.

  • If the problem involves large numbers, try with lower numbers first ($\star$)

You are being asked to simplify $7^{9999}\bmod 13$. What about $7^{20}\bmod 13$? Or $7^{20}\bmod 3$? What will happen if you multiply that by $7$, over and over? Take it down to something that you can do by hand, and look for a pattern. Having done that, if you can't find the pattern, that's fine, but now you have some material to include to your post and make it valuable.

  • Write down what you know that seems related to the problem

Any relevant theorem not in that list will be spotted right away and other users will point it out easily.

($\star$) you have to make them up yourself, and that very process is excellent to make progress in the way you think in general: what is a good, representative example in a given situation? A good example is one that is not too far off from the general case, one that gives a good idea of what is going on. Knowing your definitions also means knowing one or two good examples.

||||||
$\endgroup$
  • 5
    $\begingroup$ If the problem involves large numbers, try with lower numbers first. I'd rather change this to "If the question asks for specific cases, try some other cases." $\endgroup$ – Simply Beautiful Art Mar 11 '18 at 22:57
  • 3
    $\begingroup$ @SimplyBeautifulArt I agree that the sentence as it is now sounds very specific, but that is on purpose, I think one gets the idea like this. This is meant to be read by people with currently low/medium capacity for abstraction. $\endgroup$ – Arnaud Mortier Mar 11 '18 at 23:01
  • 8
    $\begingroup$ Totally this: "if you realise that you actually don't know the definition of one or several keywords, then you are probably asking the wrong question"... If only more people realize that they actually don't put in enough effort to make their learning journey a good one... $\endgroup$ – user21820 Mar 12 '18 at 7:31
  • $\begingroup$ Hmmm, what about difficult questions from oral exams and the like ? $\endgroup$ – Gabriel Romon Apr 13 '18 at 13:01
  • $\begingroup$ @GabrielRomon this kind of questions is usually less likely to be downvoted, and the community more enclined to discuss and give hints or ideas to be tried. At least from what I've seen. $\endgroup$ – Arnaud Mortier Apr 13 '18 at 13:25
38
$\begingroup$

Mathematical typesetting using MathJax

Mathematics.SE uses the emulation engine MathJax for providing $\TeX$-like mathematical typesetting. This means that you can use mathematical notation in your questions in a visually appealing way.

How can I use MathJax?

  • Basic information: This gives a quick start for people familiar with the $\TeX$-family of markup languages.
  • Specific information: Not every $\LaTeX$ command is supported in MathJax. For extensive documentation of commonly used constructs, see here. That thread is also good for quickly getting to grips with MathJax if you're new to $\TeX$-like typesetting.
  • Mathematical expressions in titles: The title of your question supports MathJax as well, so use it as deemed fit! However, please adhere to the specific community guidelines for their use in titles.
||||||
$\endgroup$
36
$\begingroup$

Tagging

Tags are a way to help us organize posts on this website. People also use them to locate the questions they will find the most interesting. Thus, good tagging helps to attract the best potential answerers to your question.

How do you select the best tags for your question

  • Tags are about content. Tags (except "meta" tags, see below) are supposed to refer to the content of your question, and not so much to the context in which you encountered them (but please, do add context). For example, if you have trouble solving a polynomial equation, use , even if you encountered it while solving an problem.

  • Use many tags. You can add up to five tags to your question. A combination of tags gives more information about the question, and so increases the usefulness of the tagging process.

  • Tag Wikis. Not sure if a tag fits? Move your mouse over the tag name to see its tag wiki excerpt. Read through it to see if the tag actually fits your question. Quite often the excerpts will also contain suggestions to other, more suitable tags.

  • Be wary of the meta tags. Meta tags like and do not give information about the mathematics in the question. They should be used as assistance to the tags describing content, not be the main, or only, tag to your question.

||||||
$\endgroup$
32
$\begingroup$

Formatting and writing

This is not some random internet forum. We strive for well-composed questions and answers of lasting value -- so keep in mind that your question can be of interest to others as well.

Posing a well-formatted question

  • Use proper English to the best of your ability

    The use of proper spelling, grammar and punctuation makes your question easier to understand, more appealing, and more likely to attract knowledgeable experts to answer your question. If English is not your first language and you are concerned that you will be unable to express your question clearly in English, it might be better to post your question in your native language; it is likely that another user will be able to translate it for you.

  • Make your actual question stand out

    If your question contains a lot of context, your work, or background information, others may find it difficult to figure out what exactly your question is. You can use bold text: **bold text**, a "quote":

    My question

    by > My question (on a new line), or a horizontal line: --- (preceded by an empty line) to make your question easily identifiable. See also "MarkDown formatting" below.

  • Make your question visible, searchable, accessible

    Don't force someone to click on an external link just to see your question, it should be immediately visible after clicking on your title. The key parts of your question should not be in an embedded image such as a photo/screenshot/etc., such images are not searchable, they are not accessible to those using screen readers, and they don't show up in question summaries.

  • Use paragraphs

    Nobody likes to read a densely packed monolith of text. You can enter a blank line in the editor to start a new paragraph at natural places. Add them in a natural frequency that makes for a pleasant read.

  • MarkDown formatting

    Markdown is the markup language used to format posts on this site.
    Most frequently used things are italics and bold, which you can achieve using *italics* and **bold**. To include links you can use this syntax: [Markdown](http://en.wikipedia.org/wiki/Markdown), which gives you Markdown.
    For more details and other stuff you can do using Markdown see help.

||||||
$\endgroup$
  • 4
    $\begingroup$ If I may, I would suggest possibly posting in two languages, so users who do not speak the native language of OP can try to crack the (possibly messy and incomprehensible) English during the time it takes a user who knows that language to find the question and translate it. $\endgroup$ – MickG Mar 11 '15 at 18:22
  • 3
    $\begingroup$ I would also add that key parts of the question should not be in the form of a photo/screenshot/&c. Images aren’t searchable, they’re not accessible to people using screen readers, and they don’t show up in question summaries. All too often I see questions in which the problem statement itself is a photo of a page from a textbook, while the actual text in the question amounts to “I don’t know how to do this.” $\endgroup$ – amd Apr 10 '17 at 21:41
  • 1
    $\begingroup$ I would like to add that the questioner avoid pleas. That is statements unrelated to the mathematics such as "Please help me." $\endgroup$ – Stephen Meskin Nov 28 '17 at 17:12
  • 1
    $\begingroup$ I disagree with the suggestion that headers like “My Question” be marked down with the > character. Those become <blockquote> elements in the HTML, which is semantically incorrect. Instead, I suggest using # and the like to get <h2>s (or whatever number) $\endgroup$ – Matthew Leingang Mar 20 '18 at 0:50
  • 1
    $\begingroup$ After finally encountering one too many questions consisting of not much more than an external link, I've expanded the comment of @amd into a new bullet point entry in this answer, with an added emphasis on avoiding external links. $\endgroup$ – Lee Mosher Dec 7 '19 at 16:32
6
$\begingroup$

Include source / motivation for your question

According to recent discussions on this Meta post-Is it a good idea to include source from where a question is taken? , and by the responses of many experienced users we have decided to make it into a good practice to include the source or motivation of a question.

For eg. in contest questions, citing the source definitely helps future readers, makes the thing more professional and guarantees authenticity of the question which prevents wastage of time of answerers as is seen some times due to incorrect "prove that" type questions.

Also, if someone extends theirs thoughts on something one read in a paper etc. including the source helps to understand the motivation of the user and makes it easier for others to comprehend and respond to the question appropriately. This makes the whole system more integrated, systematic and connected.

It is encouraged to willing users to help spread this idea in the main site and at times politely comment below applicable questions requesting askers to include source. Maybe like this-

This looks like a problem you have collected from / inspired by some source. According to recent discussions in Meta, we are looking forward to including sources for all applicable questions. Can you provide the source by editing the question?Refer-Is it a good idea to include source from where a question is taken?

||||||
$\endgroup$
  • 1
    $\begingroup$ Excellent! Thanks for doing this! $\endgroup$ – amWhy Oct 22 '18 at 17:10
  • $\begingroup$ @amWhy Thanks. Its an honour and pleasure to be able to contribute ;-) $\endgroup$ – tatan Oct 22 '18 at 17:14
  • 1
    $\begingroup$ With this included here, in this post, the position will get even more exposure, in addition to what it has in your original post. I also included a direct link to this suggestion in the body of the question above, so folks needn't scroll to see it. So every suggestion linking "How to ask a good question" from this post will include your "answer", and a link to it. $\endgroup$ – amWhy Oct 22 '18 at 17:18
  • $\begingroup$ This answer repeats some of the points of the "Add context" answer. I wonder if maybe we should move these to this answer, and add a link from the "Context" answer to this one? $\endgroup$ – Arnaud D. Oct 24 '18 at 12:08
  • $\begingroup$ @ArnaudD. I do not think this this specific form of context is any more important than the other forms given in the other answer. For example, self-studying a book is different from taking a course which is based on the book, and this fact is just as important as where the question is from. I would much rather a link was added to the "add context" answer directing people to the other thread, and that this answer was deleted. $\endgroup$ – user1729 Oct 27 '18 at 11:05
  • $\begingroup$ (Also, there are currently 7 answers here - it is optimistic to think that new users will read over these answers carefully and in detail before posting. For this reason I think that brevity is important, and so a simple link in the other answer is better.) $\endgroup$ – user1729 Oct 27 '18 at 11:07
  • $\begingroup$ @user1729 Please discuss with AmWhy by pinging him :-) $\endgroup$ – tatan Oct 27 '18 at 11:32
  • $\begingroup$ @amWhy See comment above $\endgroup$ – user1729 Oct 29 '18 at 10:38
  • $\begingroup$ @user1729 I think the point here is that providing a source is not being offered here as one option for providing context. That it adds to context is great. But unless someone had a sleepless night ruminating about a conjecture that popped in their head, there is almost invariably a source of the question, which ought to be cited. Even if one is ruminating about the converse of a theorem they've learned, they should cite the source where they learned the original theorem, and its statement as they encountered it, and add, "I haven't been able to find a counterexample why the converse ... $\endgroup$ – amWhy Oct 29 '18 at 10:43
  • $\begingroup$ ...ought not to hold as well. So why so we stop at 'if fie and fum, then foo.' Shouldn't it be 'Fie and fum, if and only if foo.' Maybe I'm missing something or can't yet see a counterexample to the biconditional version?" $\endgroup$ – amWhy Oct 29 '18 at 10:46
  • 4
    $\begingroup$ Sources: contest problem from "day/month/year", + "organization sponsoring competition." Textbook problem: section title, exercise, text title, authors/editors, year of publication. Class pdf of exercises: exercise number, class name, instructor, semester (year, semester). Etc. $\endgroup$ – amWhy Oct 29 '18 at 10:50
0
$\begingroup$

Avoid confusing abbreviations and notation

Try to limit the usage of abbreviations that are uncommon restricted to certain contexts, or may cause confusion. For example, not every user of the forum may know that CNN stands for Convolutional Neural Network, but could still be able to answer your question if you explained. Another case is the usage of "iff" that, on a quick read, could be mistaken with "if", thus dramatically changing the perceived meaning of the question

Something similar applies to notation. If you are going to use mathematical symbols that are rare, or have a different meaning in other contexts, please explain what they mean in your particular post. There is no shame in using plain English to explain mathematical concepts. Math notation should be a tool rather than a nuisance

||||||
$\endgroup$
  • 3
    $\begingroup$ You ask for feedback if this is a good idea. Three users vote against it, two users vocally express disagreement. Posting this, here, now, is just disingenuous. Why did you even bother raising the other question to begin with? $\endgroup$ – Asaf Karagila♦ Jul 10 '19 at 14:08
  • $\begingroup$ @AsafKaragila While opposing the idea of the meta post in question, I did suggest something like this... although I thought it would be a blurb under "Formatting and writing" with a recommendation, rather than this. Are there special rules for contributing to the solutions on this page? $\endgroup$ – rschwieb Jul 10 '19 at 16:48
  • 1
    $\begingroup$ @rschwieb: I don't think we established any protocol for adding something here. But nevertheless, it seems to me that cherrypicking the "iff" example is somehow counterproductive to the other thread. Yes, the general sentiment is good, just the execution here is to be greatly improved. (As a separate topic I think it would good if we talk about users adding content to faq questions in general and whether or not we want to have a separate editorial process before coming to the thread, similar to faq-proposed for questions.) $\endgroup$ – Asaf Karagila♦ Jul 10 '19 at 16:55

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .