Comments on the quiz questions so far:

I reviewed all of the quiz questions on-line as of this afternoon (Sunday).
If I were to make up the quiz based on what I've seen, here is what I would
have.  (I see that some new material has been added to the directory since
then; I'll try to work in comments below.)  After the proposed quiz, I have
added more detailed notes on some of the proposed questions.

==============================================================================
Proposed Quiz.

Problem 1. (Number theory)

	Solve the equation
		2x = 5 (mod 143)

	Give at least one solution mod 143; for full credit give all solutions
	modulo 143, and give an argument that you have presented all the
	solutions. [ We note that 143 = 11 * 13, a fact that you may or may
        not care to use in your answer. ]

Problem 2. (Induction)

	Prove by induction on n that 
		\[ a^n + b^n \ge (\frac{a+b}{2})^n \]
	for all nonnegative integers n, and nonnegative reals a, b.

Problem 3. (Relations)

	%% The problem from dan-1 on \equiv_m composed with \equiv_n .

Problem 4. (Posets)

	%% The "sced" problem.

Problem 5. (Proof systems)

	%% The eq-prob problem.

==============================================================================
I'm not sure if this is of appropriate length; it needs to be vetted.  There
are several proofs to do, so I think it may be about right...

==============================================================================
Comments on some of the proposed problems:

From jakov-1:
	Generally, I thought there were too many of these and they
	were too similar.  Should be simplified, if used.

	(a) Easier to find a counterexample of |A| > 3, I think.
	(b) OK problem.
	(c) OK problem.
	(d) Interesting, but perhaps too intricate.
	(e) Taking R = AxA makes this nearly the same as (a).
	(f) Taking E = AxA makes this trivial
	(g) Taking R = I make this trivial...

From dan-1:
	[recommend for quiz]
	"composite" --> "composition" in (b)
	I have funny feelings about the last sentence; we should probably
	either delete it or have them prove it.  You may have students
	trying to use this assertion as part of their proofs...

From dan:
	Problem 1:
	This problem fails unless you assume that m \neq n.
	(a) "Prove" --> "explain why"
	(b) R should have been defined in the prologue, rather than in (a),
            to make it clear that the scope of its definition includes (b).
	(c) doesn't seem to depend on (a), (b). Is this intended?

	Problem 2:
	OK...

From eq-prob:
	[ recommended for quiz ]
	(c)
	I'm a little unsure as to what is being asked for in (c)...
	explaining why there exists a formal proof ...?

From sced:
	[ recommended for quiz ]
	Why not just have a set of size m rather than size 17?
	The proof of (a) might take a bit longer to write out than one
	thinks (we need to check this), because of the "or" nature of the
	definition, in the transitivity proof.

From parry:
	Problem 1: plausible induction problem.
	Problem 2: (missing)
	Problem 3: too easy.
	Problem 4: typo at end: gcd(a,b) --> gcd(x,y)
	Problem 4: no solutions(?)

From jakov:
	(a) easy question
	(b) OK...
	(c) "describe" ???
	(d) This proof is a bit tricky, so if this problem is used
	    it is a major part of the quiz, by itself.
	(e) This is FALSE: take n = 2 and R = any nontransitive relation.
	    (Bug!)
	(f) Easy (too easy...)
	(g) same as his other sheet
	(h) This table is not clear at all. What is being asked?

From chouprobs:
	Problem 1: (binary trees)
	OK problem.

	Problem 2: 
	Make sure a and b and not both 0.
	This may be too hard for them...

From 5 problems:
	[[looking at this for the first time, I see that this is a draft
	of the quiz.  Comments below.]]

Problem 1:
	Make sure m,n are not both zero.
	(a) Too easy, since there are no solutions, which is obvious
	    once you divide by 2.
	    -- replace with my proposal ? --
	(b) This proof needs the lemma that if a|c and b|c with gcd(a,b)=1,
	    then ab|c, I think, and a fair amount of work.  I think it is
	    too hard, unless you allow them (or tell them) that they can
            use the prime factorizations of a and b and the definitions of
	    gcd and lcm in terms of those factorizations.  (This could be
            done in a set-up part, asking them to give lcm in terms of the
	    factorizations and exponents of the prime factorizations of a 
	    and b.)

Problem 2:
	[[ problems 1 and 2 together have too much weight on gcd's... ]]
	(a) OK...
	(b) Fine...

Problem 3:
	[[ recommended for quiz, comments above ]]
Problem 4:	
	[[ recommended for quiz, comments above ]]
Problem 5:
	[[ recommended for quiz, comments above ]]

==============================================================================
Final comments:
	I think we are in fair agreement about the quiz... but:
	The quiz needs an induction problem, such as the one I suggested.
	There is too much on gcd's, as noted.
	I suggest dumping the first problem from 5probs, and adding my
	induction problem (or another one of similar flavor).

	You might consider tacking on my problem 1 to the end of 5probs
	problem 2, since problem 2 as it stands is not so hard...

        The things to check are:
		(a) length
		(b) clarity
		(c) distribution of easy/hard problems (there should be
                    challenging bits for the best students, and easier bits
                    for the not so good...)
	Perhaps one or more of the graders can help "vet" them by taking it 
        tomorrow...
==============================================================================
	
Cheers,
	Ron