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Good books for logic, proofs course?

  1. Mar 3, 2005 #1

    mathwonk

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    I have to teach the "bridge" course for junior level math and math ed majors on proofs and logic, and need to find a book. I do not like books that are mathematically vacuous.

    I.e. I want one that teaches how to prove things and then actually proves something of mathematical interest, like some modular arithmetic, and hopefully also some elementary analysis (every continuous function on a closed bounded interval is bounded?). A book that spends 100 or 200 pages still puffing about sets, logic, and injective vs. surjective functions turns me off.


    I have another difficult requirement: the book should not set the poor student back 100 or 120 dollars, as several otherwise reasonable ones do (e.g. Bond and Keane). To me it is a crime that a fluffy book like that should cost more than Spivak's great calculus book.

    At the moment I am tempted to use a small, relatively inexpensive ($36), and intellectually lightweight book by Velleman, for the logic, but combined with the very substantial and cheap ($15) classic: What is Mathematics?" by Courant and Robbins for the actual mathematical content.

    I learned the propositional calculus myself in high school out of allendoerfer and oakley's excellent Principles of Mathematics. This also included complex numbers (invaluable), groups rings and fields (just the definitions, almost worthless), analytic geometry, probability, and calculus.

    Unfortunately this is out of print. And one is always afraid that a book written in the 50's may be unreadably difficult for todays average student, since it tends to assume a decent high school education, now all too rare.

    I taught the course successfully in the 70's out of Robert Stoll's book, something like Sets, Functions and Logic (which in spite of the title did prove the Bolzano Weiewrstrass theorem.) It too is apparently out of print.


    Any suggestions?
     
    Last edited: Mar 3, 2005
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  3. Mar 3, 2005 #2

    honestrosewater

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    Is this the book you mean? Also on AmazonUK.
     
  4. Mar 3, 2005 #3

    mathwonk

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    hmmm.... thanks. that fits my description but does not have the content i remember. i.e.that is all sets and logic and no math.

    either i am confused, or he wrote several books with similar titles.
     
  5. Mar 3, 2005 #4

    honestrosewater

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    From the description, "Proofs and Fundamentals" sounds like what you're looking for, but I haven't read it. $43.
    "What is Mathematics" looks cool. If you use this for the "how to prove it" part of the course, would you just need a logic book? Do you want a book that covers both logic and set theory? Symbolic logic? Mathematical logic? I've sifted through a ton of logic books; I can possibly help if that's what you need.
     
  6. Mar 3, 2005 #5

    honestrosewater

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    Okay, I think I found a really good one. This book is used in a course (course description) matching the description you gave- an introduction to higher/advanced maths. It's "Discrete Mathematics" by Norman Biggs. The first parts are about proofs and logic and such, but it also covers all kinds of interesting stuff. Lots of worked examples and exercises. It also has a companion website with Lecturer Resources. $60.
    Most of the other books I found that were used in similar courses were >$100. Another book is "Essentials of Mathematics" by Margie Hale, used in the author's course. $48.
    Both books are fully searchable online.
    Edit: BTW, you're not crazy; The book you wanted by Stoll is "Sets, Logic, and Axiomatic Theories" and is out of print. Er, I should say I think that's the book you wanted.
     
    Last edited: Mar 3, 2005
  7. Mar 3, 2005 #6

    mathwonk

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    Wow Thanks! That does look good. It has not only proofs and logic and sets and functions, but also applications to groups. I am not so interested in the applications to graphs and would prefer elementary analysis, as used in calculus, but still it seems a possible choice.

    I really appreciate your time.
     
  8. Mar 4, 2005 #7

    honestrosewater

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    My pleasure. I'm glad I can finally lend a hand to someone who has helped me so much. I'll look for some books that cover elementary analysis too. If not, maybe a nice website or online text to supplement the book you choose.

    Edit: No luck finding such a book. However, I did find, for possible supplements, both online texts/lecture notes and 3 Dover books, all less that $15.
    (These link to their Table of Contents for perusal.)
    Shilov (includes an Elementary Symbolic Logic Appendix)
    Sprecher
    Rosenlicht
    Online (I didn't read the terms of use for all of these- always a good idea):
    Fundamentals of Analysis lecture notes. These are my favorite. He sounds like an old pro (old in a good way, of course). He even includes problems. He has notes of similar quality covering several more topics here.
    All other texts and such that I found have already been compiled here, along with many more.
    I'm not sure if this is what you're looking for, but I like searching for things anyway. :smile: If you can give me a good idea of what to look for, I would be happy to try again.
     
    Last edited: Mar 4, 2005
  9. Mar 4, 2005 #8

    Galileo

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    Just a suggestion:

    Why not write a syllabus for the students yourself?

    It only has to contain what you need to teach them, it won't be expensive for
    the students (depends on you ofcourse) and you can present the material in
    the way you want to. (Make it lively and interesting for the students who are, after all, new to rigorous proofs).
     
  10. Mar 6, 2005 #9

    mathwonk

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    indeed maybe i will. over the last 5 years i have probably written over a 1,000 pages of course notes, on diff calc, integral calc, honors versions of both of those, elem abstract algebra, modular arithmetic, differential topology, basic algebraic geometry, algebraic curves, algebraic surfaces, sheaf cohomology, riemann roch theorem, and linear algebra.
    It takes a lot of time, and some students do not like the old fashioned fonts I use, but it does make it agree with the emphasis I provide in class, and I usually give them out free.

    It really does take a lot of time though. Lat time I TAUGHT THE COURSE I HAD A VERY SUCCESSFUL CLASS BECAUSE THE BOOK I CHOSE WAS SO BAD I HAD TO MAKE UP MOST OF MY OWN MATERIALS, whioch i chose from variety of places.

    Here are some of my test questions:

    II. In each case, write a different equivalent sentence, i.e. one that would have a different symbolic expression. (Hint: compare with I.)
    For example: "if you won`t help then you`ll get out of the way" is equivalent to: "you will either help or get out of the way".

    (i) If Sarah understands geometry, then she knows how to reason logically.

    (ii) It was not true that if I came to class, I would get an A.

    (iii) Dr Smith is not both lazy and stupid.

    [One student offered "he is both lazy and stupid", to which I observed that true statements are not necessarily correct answers.]

    (iv) It is not true that: either he starts class late or he ends early.

    (v) You will either finish the test or you will not eat lunch.


    IV. Write a sentence that negates the one given.
    (i) Every natural number can be factored as a product of primes.

    (ii) Every natural number which is even, is a sum of two prime numbers.

    (iii) Some politicians are both honest and hard working.

    (iv) For every man there is a woman who can love him.

    One woman, after answering correctly: "There are some men no woman can love" added: "You got that right!".
     
    Last edited: Mar 6, 2005
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