Saturday, July 2, 2016

How do you approach writing proofs?

If it's one I haven't seen before and therefore just already know how to do, then I guess my method is to write whatever I ramblingly think about it and compare to examples from the class to see how much detail the teacher wants.  Like do I have to prove commutativity or something like that.

3 comments:

  1. I've written proofs in geometry in high school, in discrete math, number theory, abstract algebra, advanced calculus, those are the ones I recall in terms of writing proofs. I also had to write a proof that 1+1=2 in the math class in tex prep 2. For that one, the teacher showed us the proof and we just had to memorize it.

    in geometry in high school, the proofs made no sense to me. they were like, flow charts or something, involving some kind of pictures, and I didn't see how they proved anything, but I memorized how to do them for the tests (and now don't remember how they worked).

    The proofs in discrete math and number theory made sense because they were just like, a series of equations or whatever.

    In abstract algebra, it seemed like we had to go a layer deeper than in discrete math and number theory. Or maybe several layers deeper. The proofs in abstract algebra didn't make sense to me because I didn't know how deep to go. I didn't know if I had to prove that addition is commutative or that equality is transitive. The teacher wanted those things to be assumed, actually, I asked, and he acted like it was obvious, but there were other things I thought were obvious that we did not get to assume for the proofs.

    The advanced calculus book started closer to the "beginning", but it said that it wasn't actually rigorously constructing the number line or whatever. But I felt like it was getting closer to explaining how proofs actually work and maybe if I completely understood that I would understand why the proofs in other classes were done at different layers.

    When going through the advanced calculus book, I did attempt to write some of the proofs out before reading to see how they did it in the book. And sometimes I would get stuck. And sometimes my proof, which I felt pretty sure was as proofy as the book could get, would be different from the book's proof. And sometimes I could see how mine and the book's were actually the same but worded slightly differently, but sometimes they seemed completely different and I didn't know if mine was right at all and I didn't understand why they were so different.

    Throughout most of the times I've had to write proofs my method has been to copy the examples given or memorize the exact proof I'm supposed to do, and then ask a teacher if what I did was right.

    I guess my method while going through the advanced calculus book, though, which sometimes was slightly different than just copying the examples, was to take the premises given (like when it says prove that a implies b, where a is a set of premises) and think of all the things I could that were also true given that and scribble it all down and see if any of it hints at b and if not then write down things that those things imply, until i see it heading towards b, or until i give up and look at the first step of the book's proof (and then try the process again from that first step).

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  2. In most classes, my "method" for writing proofs was to memorize them or copy examples from the book/class/teacher. The only class that was any different was advanced calculus. When going through the book for that class, I would try the proofs before reading the ones given in the book. The problem would be to prove that a -> b, where a and b stand in for sets of premises and conclusions. So I would write down all the premises and write down everything I could think of that those premises imply until I see something that seems to hint towards the desired conclusion(s) and then home in on those particular chains of reasoning and see if they actually lead to the required conclusion. If they don't seem to, then after a while I would look at the first step of the book's proof and then try the process over again taking the first step of the proof as another "premise". And repeat that process till I either have what I think is an accurate proof, or I end up reading the book's entire proof. In the latter case I would at least try to *follow* the book's proof. In the former case, I would compare my proof to the book's, and if they are different I would look it up online to see if other people have done it the way I did or ask the teacher (to make sure what I did was actually a correct proof).

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    Replies
    1. In most classes, my "method" for writing proofs was to memorize them or copy examples from the book/class/teacher. The only class that was any different was Advanced Calculus. When going through the book for that class, I would try the proofs before reading the ones given in the book. The problem would be to prove that a -> b, where a and b stand in for sets of premises and conclusions. So I would write down all the premises and write down everything I could think of that those premises imply until I could see something that seemed to hint towards the desired conclusion(s) and then I would home in on those particular chains of reasoning to see if they would actually lead to the required conclusion. If they didn't seem to, then after a while I would look at the first step of the book's proof and then try the process over again, taking the first step of the proof as another "premise". I would then repeat that process till I either had what I thought was an accurate proof, or I ended up reading the book's entire proof. In the latter case I would at least try to understand the book's proof. In the former case, I would compare my proof to the book's, and if they were different, I would look it up online to see if other people had done it the way I did it or ask the teacher (to make sure what I did was actually a correct proof).

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