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How to do Mathematical proof for beginner

  1. Jul 2, 2013 #1
    Hello all,

    I know that perhaps this question has cropped frequently in a lot of places, but I would like to be quite specific in my case. I've just finished High School with UK A level syllabus. The syllabus inside Math A level is mostly straight solving equation type of questions, such as for example integrating or finding root to complex polynomial. I know how take a derivative a function, to find its maxima or minima but I don't know how to do so with the so to say rigorous theorem and so on. I have to say that my Math skills in such type of question are normal, in a sense that I have to do questions carefully and cannot just simply solve one in some prodigious manners but I can grasp concepts quickly.

    Now the question is where to start with proof-based Math, the one where you need to apply definition and etc to do proof? What kind of books and what books do you recommend if you were in my position and background? I am also hoping that there's at least informal introduction or just helpful website on this matter, since most of the time I can only find sites that discusses straight questions. My target is to go and take Physics major when I enter university later on next year, and I am hoping to get as prepared as possible, especially with the maths.

    Also from looking at some examples I also want to know how do you recognize if a proof is correct? When given something to proof I often think as it's simply true, like 1+1 = 2. How else can you proof such thing, to what extent we need to prove? Previously I just need to plug in number to see if I'm solving equation correctly. In Physics you can usually do experiment or thought-experiment to see if equations or what you are doing are true, here I'm quite lost.

    Here's the syllabus for A Level Math if you are interested:
    http://www.cie.org.uk/docs/dynamic/43807.pdf [Broken]

    Thank You
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Jul 2, 2013 #2
    To get used to proofs, many (including me) would recommend "How to prove it" found here: https://www.amazon.com/How-Prove-It...1372773747&sr=8-8&keywords=mathematical+proof

    Once you get used to logical arguments, you'll get a sense to see if it's lacking anything. In math, things are usually tried to be proven in the most general case possible, so it is almost always completely symbolic, with special cases relegated so certain proofs. I wouldn't worry to much however, once you get some practice, proofs actually come very naturally.
    Last edited: Jul 2, 2013
  4. Jul 3, 2013 #3
    Thanks, I'll certainly find the book.

    Do you think also that proofing skills are essential with physicist? (and specifically the one still in undergraduate learning?)
    Last edited: Jul 3, 2013
  5. Jul 3, 2013 #4

    Exactly the book I was going to recommend. That book prepared me for my first college level abstract mathematics class. I think I only got through the first 3 or 4 chapters. The structure of it is excellent.

    Since you're self-learning, make sure you get the later edition (the one that's linked to) as the first one doesn't have any solutions, which is a bit annoying.

    Dave K
  6. Jul 3, 2013 #5
    Being able to rigorously prove something may not be AS useful with physics, but it certainly wouldn't be unusual for it to come in handy. However, being able to logically address a problem and come to rigorous solution is a rare skill these days. Just with that in mind, mathematical proof (even with very elementary theorems) can be useful for everyone.
  7. Jul 4, 2013 #6
    Proofs are the bread and butter of mathematicians. Physicists maybe not so much (except perhaps highly theoretical ones, who are often considered applied mathematicians). But there is something to be said for understanding the underlying logic to the mathematics you will be using, and the *practice* of logical thinking developed by proving. In general it teaches you to think better.

    BTW I just remembered I had this in my bookmarks:


    It's a little tutorial. Not fantastic but it will get you started.

    -Dave K
  8. Jul 4, 2013 #7
    Thanks Dave!

    So I tried one of the exercise, where one has to prove this statement: "If a divides b and a divides c then a divides b + c."


    Is my proof valid and logical? Which of the statements is the most important in this case?

    Another proof.

    Last edited: Jul 4, 2013
  9. Jul 4, 2013 #8


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    First proof: What is ##Q^+##? Do you mean ##\mathbb Z^+##? (the set of positive integers). Other than that, the proof is fine. Here's how I would do it: Let ##k_1## and ##k_2## be positive integers such that ##k_1=b/a## and ##k_2=c/a##. We have
    $$\frac{b+c}{a}=\frac b a+\frac c a=k_1+k_2\in\mathbb Z^+.$$ Second proof: When I see several lines, with one statement on each, I interpret it as saying that the statement on the first line implies the statement on the second line, which implies the statement on the third line, etc. Your calculations are fine, but if I interpret your statements the way I described, you seem to be proving that if ##a^2+b^2\geq 2ab##, then ##(a-b)^2\geq 0##. What you have written down is a great way to find the proof. But when you present the result, you should rephrase it, so that it's clear that the desired conclusion follows from the assumptions, e.g. like this:

    Let ##a,b\in\mathbb R## be arbitrary. Since the square of a real number is never negative, we have ##0\leq (a-b)^2=a^2+b^2-2ab##. This implies that ##a^2+b^2\geq 2ab##.

    I don't mean that you have to change the order of the inequalities. You can do it the way you did it if you add a comment like "the following statements are clearly equivalent".
    Last edited: Jul 4, 2013
  10. Jul 4, 2013 #9


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    So basically your second proof is also excellent, if you read the three lines in reverse order :)

    The remark about the equality is nice, but not strictly necessary for the proof.
  11. Jul 4, 2013 #10
    What Frederik said.

    A very "beginner" thing in proofs is learning to move away from just writing down calculations to writing down the "Story" of your proof in a way that is readable. See how Frederik has a few clarifying remarks in what he's written.

    What you have is good, but it's closer to the "scratch work" for your proof than the proof itself.

    Of course all this "criticism" is keeping in mind that you're just starting, so, very good!

    The Velleman book covers all of these sorts of things though too.

    -Dave K
  12. Jul 4, 2013 #11


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    I never got the point of those "learn how to prove" proof books. Why not just learn how to write proofs for a given math subject by doing the subject proper? A proof based calculus book will get you up to speed and you'll be learning rigorous calculus to boot.
  13. Jul 4, 2013 #12
    Right. Proof books are unnecessary.
    I can easily make somebody learn proofs in the matter of days. It's really a matter of getting used to the formalism.

    Of course, to learn proofs you need a good teacher. You will need to construct proofs, and the teacher will rip it apart. This is the best way to learn. But if you're self-studying and have nobody teaching you, then perhaps proof books are the only way (sadly).
  14. Jul 4, 2013 #13
    Why do you ask?

    Have you been offered a place on a Maths degree course? If so, there is probably an introductory reading list which you should follow.

    Are you considering whether to pursue a Maths degree? If so, you are already aware of a big issue: undergraduate Maths is nothing like A-level maths, and interest and aptitude in the latter does not necessarily mean that a Maths degree is right for you.

    What follows are very personal opinions:

    One book that is often recommended for pre-university is "What is Mathematics?" by R Courant and H Robbin, but I think this book is far too interesting. For a better idea of what part of a first year undergraduate course is like, try "Mathematical Analysis" by K G Binmore or "Rings, Fields and Groups" by R B J T Allenby. You will probably find these very hard going without the assistance of tutorials and a peer group, but if you find that not only can you not do any of the exercises but that you are not interested in doing any of the exercieses, maths is probably not for you. Don't worry, there are more interesting things in undergraduate maths than analysis and algebra but if you can handle these you can handle anything IMHO.

    Are you planning on another degree course, say Physics? Physicists don't do proofs, they do derivations. This is basically similar to what you have done at 'A'-level, although the topics are more advanced. If you want to get a head start on the maths behind advanced Physics I'd recommend something like "Mathematical methods for science students" by G. Stephenson.

    (And as others have said, your first proof is basically sound but your second is upside down; you need to start with the statement ## \forall a,b \in \mathbb{R} \; (a - b)^2 \ge 0 ##).
    Last edited: Jul 4, 2013
  15. Jul 4, 2013 #14


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    Proof books are not the only way, there is also learning logic and set theory. It depends on what you like, just what you need or all the details. I recommend Tarski and Suppes for this path.
  16. Jul 4, 2013 #15


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    I used to think so, but then I realized that it's probably because my first course at the university was a nice intro to naive set theory, logical notation, induction proofs and that kind of stuff. It doesn't seem like everyone gets that kind of intro. If you don't have that foundation, I think it will be too hard to learn proofs by studying something that includes a lot of proofs.
  17. Jul 4, 2013 #16
    People used to get exposed to proofs a lot earlier, like in high school geometry. This no longer happens. Now people don't get exposed to proofs often until their first class in analysis or abstract algebra. I've talked to people with Bachelor's degrees in math that said they never really learned how to write a proper proof.

    So what actually happens is that you spend some of your time learning the subject, and some of your time learning how to write a proof. Again, this used to work when people had at least some exposure to proofs at an earlier phase.

    BTW, some of us actually like mathematical logic and foundations. I enjoyed going through Velleman's book immensely - more than any math I had studied before - simply because of the logic and structure.

    -Dave K
  18. Jul 4, 2013 #17
    Note to Seydlitz:

    I asked a similar question here when I started on PF, and got a lot of the same responses along the lines of "Don't do proofs, do math." and "learn proofs through learning this subject or that subject...."

    I ignored that advice, because I wanted to do proofs even before I knew what the hell I wanted to prove. Some of us are just lead in that direction. I'm glad I did. So don't let anybody here tell you otherwise. If you want to do proofs, then do them. By the time you start learning a subject that requires you to write proofs, you'll have the formalism and language that you need and won't have to struggle like other people or retake a class because "I understood the subject but my proofs weren't good enough."

    -Dave K
  19. Jul 4, 2013 #18


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    That still happens: the proof based 9th grade geometry curriculum is still around, including in my high school. In many universities people can take honors calculus, instead of the much more trivial regular calculus, where instead of being spoon fed computational results you actually prove things; these will usually be on the level of Spivak or Apostol's calculus texts. In fact calculus texts like Spivak and Lang do a hell of a job introducing the reader to proofs in general and at least the proofs will be in the context of concentrated mathematical subjects, not spread around across random disciplines.

    The basics of proofs will not take long to pick up at face value e.g. contradiction, constructing counterexamples etc. The hard part is knowing when to use what and how to write economical proofs and for this you will only benefit from having a teacher; I fail to see how self-studying from a "learn to prove it" book will help at all with such subtleties. Luckily I first started learning how to write proper proofs (not the BS you learn in those 9th grade geometry classes) when learning topology with micromass's help and he's easily the best math teacher I've ever had so it all worked out for the better.
  20. Jul 4, 2013 #19
    Your replies have been most encouraging and helpful for me guys. I don't know where else I could get such a well-thought respond and advise, as well as experience. I will certainly strive to learn how to do this properly. It is evident that I'm not used to the formalism yet. In computational based Math, I will only usually write equations messily in order to solve something without saying anything about it, and just hoping my final answer is right. Now I know that description and order are important part of it and necessary, like for the second proof.

    No, I'm planing to take major in Physics. Though, some sense I quite like how Math is done at university level, the way you guys show it. Even perhaps more than A-Level math style of questions. In fact I used to worry a lot about what all of those definitions and theorem mean, but I found out unfortunately that it was hardly used at all during my high-school. It was all the question of, who can calculate and solve things faster in order to finish worksheet and unit tests at the given time, and also about learning the pattern of questions in past papers.

    If there had been some kind of preliminary course regarding undergrad Math here I would have certainly take it but there's none I'm afraid. Moreover, I don't live in States nor in 'West' so to say, hence the options are a bit limited, though I'm planing to continue my study in Australia next year.

    The thing is, I'm worried that what I've learned during A Levels does not correspond mostly to undergraduate study. That is why I want to learn the proofs. I want to ensure that I have good foundation in both abstract and computational Math early on so that I won't have much trouble doing the physics. I see no problem with the computational part but the abstract part is still beyond my current understanding.

    On top of that I have several months of free time until I can study in uni. (equivalent to approximately one semester of study time) I am hoping that I could spend all of the time to the maximum, lest I will be wasting my time fooling around with games and movies and be forced to struggle later.
  21. Jul 5, 2013 #20
    If your goal has more to do with physics than math, then definitely follow the advice of the posters above about going through a rigorous calculus textbook like Spivak or Apostol. You'll at least be reading good proofs if only writing mediocre ones because you will be self-studying.

    My point is that Vellemans' book (and similar) will teach you how to write *good* proofs from the beginning, because you'll get the entire structure of what a proof should look like, before you even have something to prove, and you will learn the logic behind proofs.

    Proving basic number theoretical stuff like "If a divides b and b divides c then a divides c" is a good exercise in logical and mathematical thinking. It will not help you directly with physics, but it will help you follow the logic in proof based books and eventually help you write your own.

    Do yo know what book the school you will be going to uses for calculus?

    -Dave K
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