What is your thought process as you do proofs?

[Frion]

Just wondering. I haven't been having problems with proofs, so far, but I'm interested in how people think about proofs. I feel I'm still far from ideal. There are some standard proofs, like when proving uniqueness which have all looked the same so far. There are also counting proofs, in which I am forced to think through exactly what I'm doing. But for most other proofs, I just stare at it until I get a compulsion to do something. Like factor it into a special form or subtract and add something to the expression. It's usually only by the middle of the proof that I realize what I'm doing. I suspect this is because the problems were crafted in such a way that they're easy to prove. I'm especially interested in how people come up with original proof techniques.

 

[rubrix]

translate whatever the question asks into simple words, which often involves knowing definitions of terms used in question.

Start listing what you are given (and on the side write out what you need to prove, if possible write out how the proof should end). Then use an appropriate method (induction, contraction etc) to connect the dots.

 

[TMM]

I like to work backwards, transforming what I'm supposed to prove into something more closely or easily related to the given information and then starting to work from the beginning.

 

[DukeofDuke]

I like to draw pictures whenever I can. This works well for some proofs, like anything dealing with epsilon delta's. Other things, not so much haha.

 

[snipez90]

Well I try to solve the problem and worry about writing a clean proof later. I try to get as much of an intuitive feel for the problem as I can. Working out small cases is helpful since I can get a concrete idea of why the statement should be true.

As far as general proof techniques go, I don't really resort to proof by contradiction unless I think there is a good reason a direct proof will not work out nicely.

 

[WiFO215]

Well, so far, I've been posting whatever I need to prove here on the forums, and after I go through an hour of LaTex so that I get all the formulae up, the answer suddenly jumps out at me. Once in a while, I go so far as to even post the topic and the proof hits me a while later. Magic.

 

[chiro]

I guess you could think of the whole proof like some sort of graph. Some things follow linearly from a past statement but others use multiple statements to prove a particular point.

Usually when you have a complex proof you will want to do all the trivial sub-proofs first and then introduce the main thing later on. Its kind of like writing a report in the way that proofs have structure to them just like a scientific report has structure.

Usually when I see proofs every step is usually simply a transformation of the object in the last statement to a new object in the current statement. The transformation might be taking an equation and transforming it, or a graph and transforming it, or a group and transforming it to something else in which you continually "transform" the object to something more manageable to do with what you are trying to solve. For example if you are trying to find an analytic series to something then you are probably going to start off with something and transform it slowly step by step to a series equivalent and then use known properties of series to break it down and synthesize it further.

Its almost like having a mental library of statements at your disposal and then using the right one where its needed to achieve your goal.

Good luck with it all.

 

[fluidistic]

I think intuition is a key to do proof. You get it with experience. Doing lot of proofs.
It took me about 2 years in order to be familiar with them.

 

[vortmax]

i like to use a dry erase board. This let's me draw and erase at will...more of a "stream of consciousness" approach. If you think of it, write it down. If it doesn't work, erase it. Make notes, draw graphs, draw pictures, etc. It's amazing how much the action of writing it down stimulates your thought process.

I usually end up working it forward from the start and backwards from the end...meeting in the middle somewhere. It's a very abstract flow, but I seem to work better when I can keep things in terms of my own "internal logic" until it's time to write down the formal version.

 

[irises28]

4:00pm - 4:05pm Prove the base case.

4:05pm - 12:00am Relax from the success of finishing 1/3 of the proof.

12:00am - 4:00am Panic because you can't prove induction step.

4:00am - 4:05am Find the clever trick in the induction step.

4:05am - 4:10am Conclude the proof.

4:10am - 9:00am Go to bed telling yourself that you will have a better work ethic on the following days.

-repeat

 

[Eyekana]

this is a great thread, i haven't taken upper level math classes, i'll try to remember what you guys said when i get there lol

 

[emyt]

4:00pm - 4:05pm Prove the base case.

4:05pm - 12:00am Relax from the success of finishing 1/3 of the proof.

12:00am - 4:00am Panic because you can't prove induction step.

4:00am - 4:05am Find the clever trick in the induction step.

4:05am - 4:10am Conclude the proof.

4:10am - 9:00am Go to bed telling yourself that you will have a better work ethic on the following days.

-repeat

ROFL. (Pardon my internet lingo)

That could be generalized to anything that has to do with mathematics

Something I do is think about other things that I know are true; first, you think about WHY those things are true and then you try to relate everything together

 

[General_Sax]

Mostly I just hope that my marker doesn't call me on my BS, lol.

 

[DukeofDuke]

When I don't exactly know how to do it, but I know what the picture looks like, and I need to go to my next problem set,

I use the inherent vagueness of the english language to literally "write out" my "proof" in a way that "makes sense." Yeah, not rigorous, but sometimes they buy it, as far as arguments go.