I think I have a problem. If I am asked to do a proof, no matter how simple, I need to use a pen paper. I think I'm fairly quick but if someone asks me to do the same proof in my head it takes me significantly longer. I am an undergraduate student and often in class I see people who are able to do proofs and answer questions incredibly quickly without writing anything. It is well known that there are tricks that allow you to perform computations with large numbers incredibly quickly in your head. Are there similar tricks for doing proofs?
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2Please give concrete examples, it depends largely on the proof whether it can be managed this way. – Peter Jul 09 '16 at 13:12
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Let f1 and f2 be the same function. Does the following hold: f1 (intersection) f2 iff f1=f2? This is an example I saw here as I was typing my question and it is easy to do. But if asked to do it without a pen paper it would take me a shamefully long time. – GLR Jul 09 '16 at 13:17
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The above was a very simple example. When it comes to more complex things I'm hopeless. I would like to see examples from any area of mathematics just to know that it can be done. – GLR Jul 09 '16 at 13:25
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It doesn't really matter how you do proofs, what matters is whether you can. And something is wrong in your example, if $f_1$ and $f_2$ are the same function, of course $f_1=f_2$.And $f_1\cap f_2$ (is that what you men by "(intersection)"?) doesn't really mean anything. – Henrik supports the community Jul 09 '16 at 13:27
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I think what you need is to read those legendary master's proof. And practice more try to do proof with a frame rather than focusing on detail. It will definitely make you smart. – Zau Jul 09 '16 at 13:30
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@Henrik: I assumed the question meant: if f1 and f2 are two sets with the same elements then show that their intersection is the set f such that f=f1=f2. Functions are just sets of ordered pairs. So I'm not sure what you mean when you say taking the intersection of f1 and and f2 is meaningless. – GLR Jul 09 '16 at 13:53
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@Zack Ni Part of my motivation is that if I become faster at doing proofs it will increase my chances of doing better in my exams. Part of it is just that I feel incredibly slow compared so many of classmates... – GLR Jul 09 '16 at 14:03
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Yes, $f_1\cap f_2$ is a set, but it's not anything you can but before an "iff". And the statement as written in your later comment is obvious, for any set $A$: $A\cap A=A$. – Henrik supports the community Jul 09 '16 at 14:26
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As in so many things, my recommendation is to practice.
Take a problem, try to visualize it, and see how far you can get doing it mentally. When you are stuck, write down what you have done so far, and either continue on paper or continue mentally.
The more you get accustomed to keeping ideas and chains of reasoning in your head, the better you will be.
Historical note:
Charles L. Dodgson, better know as Lewis Carroll, who was a quite good mathematician, did this often. He collected the problems, he solved which he called "pillow problems", in a book which is still available:
It is an enjoyable read, and I highly recommend it.
marty cohen
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