# List of Generalizations of Common Questions

Bill Dubuque raised an excellent point here: Coping with *abstract* duplicate questions.

I suggest we use this question as a list of the generalized questions we create.

I suggest we categorize based on topic (please edit the question). Also please feel free to suggest a better way to list these.

Also, as per Jeff's recommendation, please tag these questions as .

### Combinatoricscombinatorics

Finding a closed form for the number of strings of length $n$ over some alphabet which do not contain some forbidden substring

### Exponentiationexponentiation

Solving $x^x=y$ for $x$: Is $x^x=y$ solvable for $x$?

What is the value of $0^0$? Zero to the zero power - Is $0^0=1$?

### Group theorygroup-theory

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?

### Linear algebralinear-algebra

• Definition of Matrix Multiplication: (Maybe there should just be one canonical one?)
• On the determinant:

### Statisticsstatistics

• Upper tail inequality for the standard normal distribution:

### Topologytopology

• Book recommendations:

### Trigonometrytrigonometry

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Shouldn't the questions be "answers" rather than edits to the question? –  Arturo Magidin Mar 31 '11 at 18:27
@Arturo: Yes, we could have one answer per category. Having one question per answer might not be that good. The intent was to keep the list in one place so that we can browse through easily. –  Aryabhata Mar 31 '11 at 18:30
@Moron: Yes, I understand; but (i) we probably don't want this question to keep popping up as "unanswered". And (ii) I think if we had one answer per category (or per tag) and have people "add it to the list" there it would make more sense. Right now the question is fine, but if this keeps up, the question itself will grow too lengthy to be manageable, I think. –  Arturo Magidin Mar 31 '11 at 18:32
@Arturo: I think having this question pop up periodically is probably a good thing :-) I do agree with one tag answer thing. If enough people upvote your comment, I will make the edits :-) –  Aryabhata Mar 31 '11 at 18:37
@Arturo: I think that questions from the unanswered list only get bumped if they have an answer (necessarily none accepted or with positive vote count). (If I'm wrong, will someone please inform me?) But I agree with (ii). –  Jonas Meyer Mar 31 '11 at 18:44
@Isaac: But that might lengthen the list by a large amount. I mean both the questions in the calculus tag have other tags, namely "Integral" and "Limit," but I don't think it is a good idea to create separate categories and put them in those categories. That wouldn't be a useful addition. The geometric series is definitely in the series category, why does it need to be in algebra/pre-calculus as well? Tagging the question as such is of course a good idea. Then searching the FAQ tag on the main site is very easy. But in this list? I don't understand the need. –  Eric Naslund Mar 31 '11 at 20:04
Question: Should questions such as "Why is $0.999..=1?$" "Why is $0^0=1$? and "Why does $0!=1$" be added to this list? Should we make one question dealing with all of them at once? I mean they are asked frequently enough, occasionally with slight variations, but are definitely not "abstract generalizations." Thoughts? How could one question be added that deals with all of these type of things at once? –  Eric Naslund Mar 31 '11 at 20:35
@Jeff Atwood: I think it'd be better for each tagged subject to have a corresponding meta thread of faqs, abstract duplicates, etc. These could be displayed to low-rep users when they add a tag to a question. Better we could force them to choose a subject/tag before composing a question, so they see the subject-specific faqs first. This would go a long way towards removing the noise/overhead generated by duplicate questions (a big problem for a general-level math forum since exercises are often tweaked year-after-year to eliminate copying - yielding abstract, not exact, duplicates). –  Bill Dubuque Apr 8 '11 at 20:19
@Arturo: I am waiting for 2 more upvotes to your comment :-) –  Aryabhata Apr 8 '11 at 20:49
I don't understand why geometric series are under two categories. –  Glen Wheeler Apr 17 '11 at 19:25
math.stackexchange.com/questions/8337/… should be added, as questions about $\zeta(2)$ arise frequently. –  Asaf Karagila Apr 20 '11 at 16:02
Maybe we should add this to the list (at least until this silliness dies down and people move on to other things)? –  Ｊ. Ｍ. Apr 29 '11 at 17:52
May be this and this should be brought under the common umbrella of obvious consequences of the binomial formula? –  Jyrki Lahtonen Sep 27 '11 at 10:23
Hmm, I actually think it's a good thing that this "question" has no upvoted answers (and to that effect, please don't upvote any!). The bot will bump it up periodically, and we always get a reminder of these dupes. –  Ｊ. Ｍ. Dec 19 '11 at 0:28
@PeterT.off: Please don't forget to edit the question 1) to include the full general problem. 2) state that it is being repurposed (see other such questions from list above for a template which you can cut and paste) 3) and tag it as (faq). –  Aryabhata Apr 20 '12 at 1:22

There ought to be an entry for the / classic:

$$\lim_{n\to\infty} \left(1+\frac xn\right)^n = \exp x$$

I found these posts on it so far:

which I've ordered from most to least suited for status (after a suitable amount of editing). There are probably more, but I couldn't find them by searching the site.

(I've posted this answer because it is not obvious which one of the above should get status.)

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You can add it yourself. Please feel free to do so. Pick one as the representative, and tag that as FAQ. I also downvoted this answer so that this question is periodically bumped up (no answer with > 0 votes). –  Aryabhata Jan 7 at 23:07

The Lindemann–Weierstrass theorem can be used to prove many numbers transcendental, as in

There are some other relevant theorems, like Deciding whether $2^{\sqrt2}$ is irrational/transcendental.

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Either I have misunderstood you, or you seem to be missing the point. –  Aryabhata Nov 26 '12 at 6:02

This "answer" is simply suggesting another category (Logic? Proof-writing?), perhaps each a separate category) for frequently asked questions about

Logic:

• the meaning(s) of implication (and understanding it, intuitively),
• understanding quantifiers in predicate logic; ($\exists$ vs. $\forall$),
• equivalence between an implication and its contrapositive,
• understanding what is meant by a tautolology,...a contradition...etc.

Proof writing:

• understanding the difference between proving the contrapositive of an implication and proof by contradiction.

• understanding the "logic"/and or intuition of proof by induction, and when it is appropriate to use.

• Proof by induction: How to prove $P(n)$ by induction on n.

This "post" is merely a rough draft. I can certainly work at it, updating to include representative links, and revise this answer for inclusion as a category/categories in the original post, at top.

But I invite anyone noticing the frequency of such question and/or conceptual confusions to suggest links to representative posts, with an eye for the quality of ensuing answers in those post. Also feel free to include any other frequently encountered questions tagged "logic" and/or "proof-writing"/proof-strategies that I have not yet addressed in this post.

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You might just want to go ahead and add the relevant questions in the OP... –  Ｊ. Ｍ. Nov 15 '12 at 2:57
@J.M. I'll work on it; I just wanted to invite suggestions. And this also gives me the chance to use this answer as a "sandbox" prior to modifying the actual post above. –  amWhy Nov 16 '12 at 14:39
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Let's try to keep these in order by post ID number so that it is easy to avoid duplicates. –  MJD Apr 8 at 2:39
How would I create a tag "Polya/Burnside" on MSE and if possible re-tag these questions automatically? –  Marko Riedel Apr 28 at 23:26
@MarkoRiedel Tags are created simply by adding the new tag to a question. When creating tag, it is good to write also a tag-wiki/tag-excerpt for the new tag. It's up to you whether you discuss creating the new tag on meta first, or whether you create tag without such a discussion. But I think that it is a good thing that those questions will be bumped when retagged - other users will be alerted about the new tag by this and if they disagree, they can react on meta. –  Martin Sleziak May 6 at 8:07
This is one topic (Polya-Burnside) rather than several. The questions on coloring cubes and counting necklaces get asked all the time over and over again. That's why MJD suggested we make this list. –  Marko Riedel Jun 28 at 13:53

My suggestion is that there is an "examples" tag which can be used to link/merge examples of general questions into the most general question which has been posted on a topic.

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Presumably this needs to be disambiguated from the (examples-counterexamples) tag used for questions actually asking for examples and counterexamples. –  Willie Wong Oct 19 '11 at 13:48

Is the goal to find examples "in the wild" of common questions, that we then coopt and retag as faq? Or do we create new community wiki questions tagged faq specifically for the purpose of answering them comprehensively (as was done on SO)?

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The former. And we usually don't bother until at least a couple dupes appears. –  Willie Wong Nov 18 '11 at 12:50
Mostly the former, as Willie indicated, but sometimes the latter when there have been numerous specific examples of some nicer general principle, none of which would have been easy to edit into the canonical question. –  Isaac Nov 19 '11 at 15:58