Sec. 7, Due March 30, 2012

Of what we've studied for this test, the three most important things are the Schroder Bernstein theorem, the division algorithm and the fundamental theorem of arithmetic.  What I most need to review is problems dealing with the cardinalites of sets.  From this course the most valuable thing I've learned are the techniques necessary to prove something, which will surely be helpful in future math classes.

Sec. 7, 11.6-11.7, due March 28

From these sections I didn't quite understand the proof that shows there are an infinite number of primes.  I understand how they got there contradiction, but not how they saw how to reach it.  Section 11.7 had some cool things to point out, but that was about it.  It said, "Hey, this is cool and interesting, but we've got nothing else to do with it."

Sec. 7, 11.5, due March 26

From this section the only proof that confused me was the one for corollary 11.15.  They are using induction.  It says Now let a1,a2,...,ak+1 be k+1 integers, where p|a1,a2,...ak+1.  It seems as if they skipped a step here.  It seems as if they assume the statement is true for k+1 rather than k.

Sec. 7, 11.3-11.4 due March 23

At the moment I'm rather sleep deprived, so not very much of the sections made much sense.  Finding the GCD was the most confusing thing for today.  The proof puzzled me, but the application thereof made enough sense.  Hopefully I understand better in class.

Sec. 7, 11.1-11.2, Due March 21

The proof for the division algorithm was slightly confusing, but the existence of such a thing does seem logical.  It's nice to be moving back to more familiar things though.  If there is enough class time further review of chapter 10 would be appreciated.  Thus far chapter 11 doesn't seem to horrible.

Sec. 7, 10.5, Due March 19th

Many of the proofs from the section are rather difficult to follow.  They seem convoluted, but the theorem's they provide seem as if they should be useful and easy to apply.  The Axiom of Choice in particular appears to be a powerful tool for proofs.  Hopefully I'm able to reach a more complete understanding of the proofs through lecture.

Sec. 7, 10.4, Due March 16, 2012

As we've already covered this in class there weren't as many potentially confusing thing.  The proof that the power set of A and 2^A are numerically equivalent still doesn't quite make sense though.  That and the proof showing that a bijection doesn't exist between the set A and it's power set.  If we're still on this section it would be helpful to see these again.

Sec. 7, last half of 10.3, due March 14, 2012

The rest of the section appears to have already been covered in class.  It was taken up with the examples proving that (0,1) was numerically equivalent to the set of real numbers.  The main thing that would be beneficial would be to do as many examples as possible during class time with proofs typical of sections 10.2 and 10.3.

Sec. 7, first half of 10.3, due March 12

The main thing I can't understand is the proof by contradiction used to show that (0,1) is an uncountable set.  Somehow they took a1=0.31717..., a2=0.1515..., and a3=.04000....  Because of this it said that the first three digits in the decimal expansion of b are 5, 4, 5, or b=0.545.  Anyway, more explanation of the proof would be helpful.  Once you're able to assume that theorem it should become significantly easier to prove that something is uncountable.

Sec. 7 10.1-10.2, Due March 9

In section 10.2 the main thing I didn't quite understand was the tables that were used to assist in several of the proofs.  Particularly in the case of when proving that Q+ was a denumerable set.  In order for the function they created to be bijective they decided to ignore any numbers within Q+ that would have had the same value so that it would be one to one.

Sec. 7, exam study, Due March 5.

Of the topics covered I think that of mathematical induction is the most important.  Tying for second are equivalence relations and bijective functions.  I expect to see questions on the midterm that test for knowledge of the concepts given in class.  One should be able to be fairly successful just through knowledge of the definitions and application of the theorems/concepts in ch. 6-9.  For the exam I think what I need to review the most is the strong principle of mathematical induction, as well as proof by minimum counter example.  I would like to see more examples of these sort of problems during class.

Sec. 7, 9.6-9.7 Due March 2

The sections remind me of some of my high school math classes.  Inverse functions and permutations.  In the section on inverse functions the example of ln(x) and e^x were given.  It states that they are the inverses of one another, but doesn't say how you would go about proving that.  The only method it gave for finding an inverse was swapping x and y in the function and then solving for y.  As the book states that doesn't work in all cases.  What else can be done to prove that functions are inverses of one another?