When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?
Often times I "feel" as if I can write a proof to an exercise but most of those times I do not feel confident that the proof that I am thinking of is good enough or even correct at all. I can sort of think a proof in my head, but am not confident this is a correct proof.
Any input would be appreciated. Thanks.
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1) This is your teacher's job to check your proofs. He is experienced and trained to read proofs and determine what is acceptable and suitable for your level (is it safe to ignore some minor flaw? How detailed a computation should be?...) Don't hesitate to ask advice on proof writing, including on non required work and self-study.
2) Do a step-by-step verification: are all formulas correct? Are all equivalences really equivalences? In particular, don't be lazy in that step: really check that the equivalences you used are not implications. Also, check carefully all the conditions before applying a theorem: the Alternating Series Test requires a decreasing sequence? check it! Including obvious conditions: if it is obvious, write it in one or two lines. As a teacher, it always upset me when students complain about their grade for an obvious fact they did not bother to state.
3) In order to help the verification in step 2, take some writing habits: for example, introduce the notations for the objects you're working on, don't write equivalences, only implications (i.e. to prove $P \Leftrightarrow Q$, prove $P\Rightarrow Q$ and then $Q\Rightarrow P$); prove equality of sets by double inclusion; a proposition starts by "for any $x$ in $A$,...", then start writing your proof by "Let's $x$ in $A$. Then...". Also, even if math is not high level litterature, good writing skills are absolutely necessary: especially, logic connectors like "if", "then", "so", "therefore", "but", "since",... have a precise meaning. Be sure you use them properly, since IMHO they greatly help to structure a proof and keep things clear.
4) Finally, math is not divided into two steps, one when you receive a lecture with definitions and proofs from the teacher like a sacred text, and a step where you do homeworks and try to copy the master. To be critical on your own work, you have to be critical on others'work. Efforce yourself to be question the professor's proofs: why did he introduce this? Can we shorten the proof like that? He used a non-intuitive trick in the proof; can we do it without the trick?
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If you fully grasp logic, and can justify every single step you take without any 'hand-waving', you can be quite sure that you didn't make a mistake except perhaps for careless mistakes.
Unfortunately, many people I've met don't fully understand the logical structure of proofs, and that impairs their understanding of every topic, so it's the first thing to make sure you can understand perfectly. To give some examples, you must be able to identify precisely what statements have been derived and what have not, and exactly what assumptions form the scope of each statement.
The next part involving justifying each step in the proof requires you to be able to follow rules strictly. Although intuition is a very good helper in finding the proof, it is not so reliable in checking the proof, and we often have to stick to symbolic manipulation according to rules if we want to be certain of its correctness. If we don't really know what exactly is allowed and what is not, it is time to look very closely at the precise rules and the justifications behind them.
The last thing about carelessness can't be avoided without a computer, however, and the only other alternative besides a formal proof assistant like Mizar is to let someone else see your proof, preferably someone who meets the first two criteria above.
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Assuming you really care about qualifying your proof to your self before you present it to others, I would think the answer is really simple (and a bit annoying :) ); retrace your entire proof, step by step. Although there is a caveat, you shouldn't be actually attempting to solve the problem 'anew'.
Be a bit of a nag. Test, test and retest.
My math skills aren't the greatest so I'll just talk about how I'd work through testing my own proof.
I would lay out my assumptions clearly; maybe look through my solution and write down all the rules and tricks I used. Then I would check if any of my assumptions introduce ideas I didnt really think I was using. This can be even harder but, having a good reference to the topic helps.
Then I would go through the motions of solving the problem exactly as I had already done and look carefully for errors in my algebra ("Wait! What? Where did I get an exponent from?" kind of errors.)
During my undergraduate course, the Head/Dean of my math department tried very hard to teach me how to rewrite the entire solution as a sequence of logical statements. Sadly, I did not learn all that well. However, as one of the answers above points out, logical statements can be tested (easily enough if you follow the rules) and therefore are something you can work with!
Finally, the one thing that counts above all the advice is that the easiest way to solve something is to solve a bunch of easy problems like yours and then apply the same tricks/rules. If you aren't inventing crazy rules to solve your problem, you are very likely correct. However, if after repeated testing, you think your proof is consistent, that is when you want someone with more experience to take a look. Get a peer-review. :)
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In many proofs, the hard thing is to find the way which path to follow; once that is done they are easy. Faced with a problem, most of the time my result is: "I have no clue how to solve this", "I have a start but I'm stuck at some point", or "I have a proof which is correct unless I made a stupid mistake".
If you have no confidence that your proof is correct apart from possible mistakes, then you likely don't have a proof. If you say "A => B because I say so" in your proof, especially if you say "A => B must be true because otherwise my proof doesn't work", then most likely you don't have a proof. If that isn't the case, then most likely it's just a matter of checking your proof for mistakes.
Anyway, there are many exercises, too many to do them all. To learn, it isn't necessary to do all the proofs to the last excruciating detail; it's enough to get to the point where you can say "if I spent another hour or two then my proof would be faultless". Training your brain to get the right ideas so you can find proofs is the important thing. For new results, you want faultless proofs (and to avoid failing a test :-) For exercises, someone has written a faultless proof at some time.
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Use a computer with automated proof checking software, also called a proof assistant or interactive theorem prover. Typically you will need to write your proof in a special, machine readable format (be careful for translation/copy errors), but past that point this field is well studied and computer-based proof checking is generally reliable. Wikipedia has a comparison table of different software for this purpose: http://en.wikipedia.org/wiki/Proof_assistant . (If this is preferable to, easier than, or even faster than hand-verifying your proof step-by-step is another question.)
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"Proof" is an argument, that strictly follows logical inferences. That is all there is to it.
Do not be intimidated by the term "logical inferences". If you have trouble with them, practice on something like Sudoku.
In my opinion, Sudoku puzzles are great because they help introduce people to the concept of "proof". Try a really hard Sudoku puzzle. In the middle, you don't know the whole solution yet, but, you may be able to tell that this particular square "must" contain so-and-so number, so you can write it in. You did not merely "try" this number, you wrote it in for sure. How did you do that? Well, you went through some precise argument. That was a "proof".
Ask a more experienced person. IMHO that's really the only option, and one of the reasons for this is that it is very important for a proof to communicate a result and its justification to another person. If the proof is good enough to convince yourself, that's a start, but the real test is whether you can express it in such a way as to convince someone else.
And BTW... the same applies if the textbook does have a solutions manual. Your proof is inevitably going to be different from the one in the book, and it takes a lot of experience and mathematical understanding to decide whether the differences are important or not.