Andy, 

     I read/skimmed the draft "Introduction to Proofs" today.
  I'm happy to use it for the new MATH 243 this coming fall. 
  I do have a few comments/suggestions.  Hope this is useful.

                                            John 


  Chapter 1 -- looks good      

  Chapter 2 --  
 
  (a)  At start of section 2.3, I found myself wanting more
       about what proofs are and why we do them (maybe a 
       paragraph's worth -- similar to what you said later on page
       37).  

  (b)    I like the "Preliminary Work" and "Written Solution" headings.
       in these first examples.

  (c)  Example 2.28 -- first time "if and only if" occurs
       in a "real" statement (could probably use a backward
       reference to page 10

  (d)  page 24:  "putative conclusion" is OK, but (sadly), I 
       doubt too many students know what it means.  How about
       "claimed conclusion?"

  (e)  page 25:  Another good piece of advice: Read your 
       solution out loud to yourself or someone else.  (Your
       internal "BS-meter" can often help you catch statements 
       you are not sure of.)  

  (f)  page 27:  Exercise 2.13.  Don't you want to say that 
       b = a is the hypothesis and then the reason for 
       b^2 = ab is that you multiplied both sides by b?

  Chapter 3 -- maybe add "and Integers" to the title?

  (a)  I don't especially like the idea of calling 3.1 (the Peano
       Axioms) a "Theorem" here.  I know you give the usual 
       set-theoretic model that satisfies those axioms in Section 3.4,
       and that is good. But I would prefer to keep the distinction
       between axioms as unproved starting assumptions and
       theorems as things you prove.  Plus, whether or not there
       is a model that satisfies a collection of axioms is a
       separate question.  

   (b) page 34, line -4:  typo "destributive"

   (c) page 38: the short paragraph before Example 3.25 is 
       subtle and might be confusing ("you just said k is 
       particular but arbitrary -- doesn't that mean k is 
       arbitrary?").  I think the "particular but arbitrary" 
       formulation in the preivous sentence is probably enough 
       at this point.  

   (d) page 43: Def. 3.39 -- a pedantic point, you defined what 
       what it means for a mapping to be bijective, but not what
       a bijection is :)

   (e) footnote on page 44 -- lol! 

   (f) page 44:  Definition 3.44 -- another pedantic point here:
       you are identifying the mapping f with a element of the 
       m-fold Cartesian product in Example 3.47 which is essentially
       harmless, but possibly confusing.  ("Didn't you say an
       ordered m-set was a mapping?")

   (g) page 45:  proof of Prop 3.48 refers to Prop 3.48(!)  Did
       you mean 3.36?

    Chapter 4 -- 
 
    (a) Remark 4.9:  " ... while one might expect gives 
        −30 = -2 11 - 8 ... "  How about "while one might
        expect it to give ... "

    (b) I think it would help to move Example 4.22 after
        Definition 4.23 and include the observation that
        gcd(12,18) = 6, according to the definition, adding
        a short explanation why "greatest" also agrees with
        greatest in terms of the ordering on Z.

    (c) page 67:  shouldn't the comment on the while say
        while b DOES divide a?  Also this really relies on
        syntax/semantics of a particular programming language  
        more than I would like to see in pseudocode if we 
        include it.  I would be happier with something like this:

        while b <> 0 do
           rem:=remainder(a,b);
           a := b;
           b := rem;
        return a

        It would also help to point out that the value 
        returned is the last nonzero remainder. 

    Chapter 6 -- 

    (a) page 82:  typo (extra "in") in "usually the cat in a in 
        cat-and-mouse conflict"
    
    (b) page 95:  I think the footnote about security (or not)
        of RSA might make more sense to a reader if you saved
        it until after the system has been introduced (say until
        page 99)

    Chapter 9 -- 

    (a) Part of me wants to say this is coming very late in the 
        book.  You already introduced some of the ideas more
        informally in Chapter 3.  Maybe this is OK, since you
        are "spiraling back" to expand topics introduced before, 
        doing more with mappings on domains that are subsets
        of R, etc. 

    (b) p. 155 -- first use of the Axiom of Choice, I think?!
        It might be worthwhile to say that this is intuitively
        clear, but it is not as trivial as it sounds(!)

    (c) It would be nice to show that congruence mod n is
        an equivalence relation on Z to link up with topics
        from Chapter 6.

    Chapter 10 -- 

    (a) The two series in the intro are so similar that I'm 
        wondering whether we need both of them(!)  A different
        convergent series, say the sum for exp(1) might be 
        more interesting?

    (b) Is anything really gained by phrasing section 10.2
        in terms of complex sequences?  Almost all of the examples
        are sequences of real numbers and the results about 
        monotone sequences and convergence only make sense in
        the real case.   Some users (e.g. me, 
        maybe) might not want to do some much with C before this.  

    (c) This proof of Bolzano-Weierstrass is REALLY NICE!

    (d) Theorem 10.79 (Ratio Test) -- why not say that there
        is no conclusion of rho = 1?

    (e) Exercise 10.23 -- why not add a part showing that 
        the Cantor set has an surjection to the interval
        [0,1] by mapping the number defined by the sequence
        a_k to the number with base-2 digits b_i = a_i/2.
        Could also ask if that mapping is injective.

     Chapter 11 -- I doubt we would have time for this in 
     MATH 243 and it's covered in MATH 244 in any case.

     Chapters 13 - 16 -- I don't think we'll want to do very
     much (if any) of this, since it will be covered in 
     the same or greater depth in Modern Algebra