As someone entering graduate school this fall, this is something I would like to know.

For undergraduates, I know that professors generally want students to show very rigorously and clearly their chain of reason when writing proofs. This generally includes showing things that are "obviously" true.

How is proof writing generally done at the graduate level for homework and tests? On one hand, I don't want to be too terse. At best I could come across as arrogant and at worst points could be deducted.

On the other hand, showing too much could be bad. It would waste valuable time on an exam, and the professor's time grading it. It may also come across as pedantic.

So is there a rule of thumb of what can be said to be "clearly" true? Or would it be very specific to both the course and the instructor?

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    $\begingroup$ If you're expected to know it, and it's true, you can probably just mention the concept without presenting the proof. That said, I have had substantial test score disagreements with professors over this. The best thing to do is consult with your professors regarding their expectations. $\endgroup$ – Emily Jun 5 '14 at 16:32
  • $\begingroup$ Also, don't use "clearly" or "obviously." If you find yourself falling upon one of those words, stop, gather your thoughts, and be sure. It is the fastest way to a wrong proof. $\endgroup$ – Emily Jun 5 '14 at 16:33
  • $\begingroup$ I'd say it is 100% dependent on the professor. Typically things that are proven in class can be considered proven and done, but professors are people and people can be cray-cray. $\endgroup$ – Carser Jun 6 '14 at 4:08

Definition of "clearly true" depends a lot on the professor. I'd say there's no hard and fast rule of what changes when you go to grad school. Some facts do not have to be proven anymore, but other facts have to be treated more rigorously.

I remember being shocked at first getting a C from a prof who was really a neat-picker. I did adjust eventually.

I would start a bit conservatively (meaning writing a bit more down) but wait for the feedback from the prof. Better yet, talk to him/her in the beginning and try to see what his/her particular demands are. Maybe show him/her samples. Usually profs like being talked to.

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    $\begingroup$ Definitely talk to them! Although I'm an undergraduate student, there are numerous times when I've done proofs and skimmed over certain assumptions and still received full credit by virtue of the fact that my professor understands I'm aware of them (because more often than not I'll nitpick the proofs he gives us in class for not specifically identifying certain presumptions made). $\endgroup$ – Pockets Jun 5 '14 at 16:42

I have no experience whatsoever as a math student since high school, so feel free to downvote this answer or whatever.

I would look at papers the professor has written, especially papers co-authored with other mathematicians. Aren't university professors supposed to publish or perish? Maybe some of the topics will be way over your head. But I would look for phrases like "it is easy to see" or "it is obvious that" such and such.

Lastly, and this is just me guessing, anything you could reasonably expect a decent high school student to know can probably be omitted from a proof at the university level.

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    $\begingroup$ Most published papers aren't really intended for first year grad students or people not familiar with the field, so it isn't a good indication. $\endgroup$ – Batman Jun 6 '14 at 4:04
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    $\begingroup$ Published papers are written to a completely different standard compared to the grad school homework, exams etc. Think about a PhD thesis (usually near 100 pages) and papers from it (maybe 3-4 times shorter). You don't prove most well-known things in a published paper, but you might be expected to in a grad student paper. $\endgroup$ – PA6OTA Jun 8 '14 at 15:31

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