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Transition to Proof Writing

  1. Jun 2, 2008 #1
    Hi, I am looking for some opinions on making the transition from computations to proof-writing. Next year, I plan on taking Honors Calculus at the University of Chicago and rather than learning axioms or any new mathematical concepts, I'd like to learn how to express mathematical ideas better. For the past few weeks, I have been trying to prove many of the theorems that I only used, but usually didn't prove in Calculus BC. But doing proofs is a bit time consuming and getting others to check it is not an easy option for me.

    I have multiple calculus textbooks, problem-solving books, and How to Prove It by Velleman. At the moment, I am considering reading various proofs in my calc textbooks and then doing more proof-based problems to see how I have improved OR read through How to Prove It. I don't think How to Prove It comes with a solution manual, so I guess the question is, should I go with a direct approach by looking at proofs or a more "formal" training in proof-writing? Any suggestions, especially ones that I haven't suggested, are welcome.
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  3. Jun 2, 2008 #2


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    Just read and write as many proofs as possible. Proof writing isn't something that can really be taught. It's a skill that has to be practiced and sharpened over time.
  4. Jun 2, 2008 #3
    The book I used in the silly proof class that I took is not wonderful, so I won't rec it, but I'll tell you what you should look out for--

    (1) Logic: truth tables, tautologies, deMorgan's Laws etc, distinguishing between "if, then" and "iff", or vs exclusive or, contrapositive
    (2) Set Theory: usual jargon plus how to prove set equality
    (3) Numbers: equivalence relations and classes, induction, complete induction, pigeonhole principle
    (4) Structure: axioms vs definitions vs theorems
    (5) Proof methods: induction, contradiction, construction, contrapositive, counterexample

    You can learn the material in perhaps just a few hours but a solid few weeks worth of exercises in a good textbook will do wonders. It's worth the effort to learn these before diving into a serious pure math class simply so that you are only struggling with understanding the logical structure and not the language and methods used to convey it.
  5. Jun 2, 2008 #4
    yea david pretty much hit it on the head. although i would order them like this (1),(5),(4),(2)(3)

    i would also add functions,cartesian products, finite and infinite sets. basically start with truth tables and how to prove an if then statement then logical equivalence then by contradiction and then by induction. after that you'll be set to actual write proofs and you can use the rest of the material to practice. ask questions on here, we'll be glad to help you.
  6. Jun 2, 2008 #5
    Velleman is an excellent text book. If you have the second edition, there are solutions to a few problems. At the very least get through the first 3 chapters - this is everything you need to know about proofs. And you will be glad when you start that you had. Most of them aren't too bad, and you should be able to go through them just by studying the examples. Its all the stuff the other guy listed and more.

    As for learning to write proofs, they will be very time consuming - especially at first. My advice is you figure out what text book is used in your honors class, and work the first bits and take your time. If its something like Spivak, you can order the book early and get a solutions manual from amazon... you can work the first 2 chapters which would sharpen your proof skills.

    The best way is to practice. I personally learned proofs from Velleman and an elementary Linear Algebra book. Look on the internet for things like proof of area of circle, pythogoras's theorem, vectors, etc. Another good book is What is Mathematics by Courant. It doesn't teach proofs explicitly, but it goes over a lot of the theory and proofs of elementary mathematics. You can probably learn a lot from it in terms of seeing proofs, but there are no solutions to the exercises.

    If you want, I can e-mail you some of my problem sets which had elementary mathematics (eg. proving a number is rational). The beauty is I have full typed up professor solutions, so it would be an excellent way for you to see how a proof is done. Leave me your e-mail via PM.
    Last edited: Jun 2, 2008
  7. Jun 2, 2008 #6


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    try learning to write english essays well. i find many of my scholars in writing proofs fail to include verbs in their sentences, which i find makes their arguments signally less cogent.
  8. Jun 2, 2008 #7
    Hey, Howers, would you mind emailing me those problem sets as well? I'm currently preparing for all my proof based upper-division courses and could use all the practice I can get.
  9. Jun 3, 2008 #8
    Yea I dont care, PM your email or leave it here.
  10. Jun 3, 2008 #9
    Thanks for all the replies. It looks as though there's no one right way to learn proof writing, so I'll try to take in as many of these suggestions as possible. David and ice, How to Prove It covers most of those topics so I'll be sure to read through that.

    Howers, Spivak is the textbook used in Honors Calculus, and the first 2 chapters sounds like a great way to sharpen proof-writing without getting ahead of myself. I think 5 chapters in How to Prove It is enough and I certainly want to spend more time practicing proofs.

    Mathwonk, I have taken AP English and I know exactly what you're getting at. My essays are decent and the grammar is not a problem, but it takes me very long to express myself clearly. I plan on reading through Elements of Style, which is supposed to be a very good book on composition.

    I'll be sure to post any questions I have here, thanks for the help.
  11. Jun 3, 2008 #10
    Okay, I just want to say that Spivak is coming out with a fourth edition sometime this month... so you should delay on buying until maybe mid july / august. Really, all the editions are practically the same but if your school is using the newest version you might want the current edition. But our math department still uses the 2nd edition, which is over 20 years old. In the meantime, you can borrow an older version from the library if possible.

    Spivak will sharpen your skills but will not teach proofs. If you do not know how to write a proof, you won't get far in Spivak. This is why you should have Velleman down before you attempt Spivak. And this is why you should buy the answer book for Spivak, called "Answer Book".
  12. Jun 3, 2008 #11
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