Sec. 7, due April 11, 2012

Of the theorems and proof techniques from this semester I would place as the most important induction, those  theorems dealing with the cardinality of sets.  What I need to work on most for this exam are those things covered earlier in the semester that I've probably began to forget.  A problem I would like to see is on dealing with a proof by minimum counterexample.  The most important thing I've learned from this course are the techniques to creating a proof, and the way of thinking that goes with it.  If I go further in mathematics these will certainly be helpful in later classes.

Sec. 7, 12.4-12.5, due April 9, 2012

Continuity is something familiar from calculus classes of the past.  However section 12.4 offered some new properties of limits that I've not seen before.  They make sense though and should offer as little trouble as the sections preceding it did.

Sec. 7, 12.3, due April 6, 2012

When I was in calculus epsilon delta proofs were the bane of my existence.  Hopefully this time around I can understand them more.  Just reading the chapter seems to have already helped with that.  The proofs seem fairly similar to what we were doing in the last section, the main difference being defining some delta.

Sec. 7 12.1, Due April 4

This section is somewhat more familiar.  The trouble will come from not being able to simply take the limit of the sequence as has been taught in calculus classes.  Instead I'll have to prove that the limit is indeed what i find it to be.  It doesn't appear overly difficult though.

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?

Sec. 7, 9.5, Due February 29

This is somewhat more familiar territory in mathematics.  I'm assuming most students in the class have also had prior experience.  The trick here is remembering if you write the composition of  functions as f(g(x)) or as g(f(x)).  Once I can keep it straight I should be okay.

Sec. 7, 9.3-9.4, Due February 27

One to one and onto functions seem easy enough.  The main difficulty I foresee myself having is with onto functions.  The example in the reading that made this most clear was a function from the integers to the integers defined by 2n.  For a while I did not see why it was not considered an onto function, but then it became clear that it was due to the domain, and so the output of the function was limited to the even  integers.

Sec. 7, 9.1-9.2, Due February 24

Defining a function as the book has doesn't seem to different from what I've learned in previous math classes.  The main difference is simply in the notation (or so it currently appears).  The sections introduced a lot of new vocabulary concerning functions.  Things such as "mapping"  and "image".  The set B is referred to as the codomain as well as the range of a function.  Are they interchangeable?

Sec. 7, 8.5-8.6, due January 22

I found multiplication and addition to be well explained.  The comparison to a clock worked well.  In section 8.5 the book said that congruence mod n would yield n equivalence classes, then in the very last example there exist only two equivalence classes.  It was explained briefly, but I would appreciate a little more understanding on the example.

Sec. 7, 8.3-8.4, Due February 21

The main difficulty I foresee with this section is defining the equivalence classes.  It was easy enough with specific examples, it only became more difficult when it was a more general case.  I would appreciate a little more time spent on these during class.

Sec. 7, 8.1-8.2, Due February 17

The only confusing thing about the section was the books insistence on using R rather than the actual mathematical symbol that it represented.  It gave all of the relations the same symbol, and when multiple examples were given in the same paragraph it became ridiculous.  Hopefully we get past the usage of R quickly and to being able to just use the symbols.  I'm curious if the properties of the relations (symmetry, transitive, reflexive) will actually be helpful in future proofs.

Sec. 7, 7.1-7.3 Due February 15

There seems to be nothing new here.  The main thing is deciding if a proof is necessary or if the statement is false.  The authors seemed to enjoy making the section that revisited quantified statements as convoluted as possible.  The statements contained within were slightly confusing at first, but only required that I read slower.

Sec. 7, 6.3-6.4, due February 13

The main thing that I found difficult was the strong principle of mathematical induction.  They took some integer i such that m<=i<=k and then never really seemed to use it in the proof.  It also seems as if finding out the minimum value of k to be used in the proofs was obtained by guessing and then checking.  Hopefully there is a faster way than having to run through the first few values allowed by the domain.

Sec. 7, 6.2 due February 10

The idea behind the section seems easy enough.  Rather than making the base case n=1 we can use some m.  Some of the examples however still managed to confuse me, namely the one involving De Morgan's laws.  It seemed as if they started out assuming what they were trying to prove.

Sec. 7, 6.1 due February 6

For section 6.1 I understood the algebra that the book went through, though it was unclear to me why they sometimes did what they did in pursuit of the proof.
Of everything we've studied thus far, the most important things are probably the proof methods: direct, by contrapositive, by contradiction.
On the exam I'm expecting similiar questions to those we've been seeing in he homework.  I don't know which way they'll go on difficulty.  Possibly easier as we will not have the assistance of any notes or T.A.'s.  I'm expecting a few things from truth tables, set theory and what not, but the meat of the exam should be proofs.
The main thing I need to work on for this exam is proofs involving sets.  I know the general idea behind it, I just need more practice.

Sec. 7, 5.4-5.5 Due February 3

For the most part I was able to understand existence proofs.  The one example that was slightly confusing to me was the one that stated, "The exists irrational numbers a and b such that a^b is rational."  Disproving statements seems easy enough.  It essentially seems as if we're doing normal proofs with the intent of showing something is not true rather than true.

Sec. 7, 5.2-5.3 Due February 1

The proof by contradiction seems straight forward enough.  I particularly enjoyed the illustrations that look as if it was the author who drew them.  The story problem that went with the pictures was slightly confusing though.  The main thing I need to get this down is to simply practice it.  The review of proof methods was also helpful.

Sec. 7, 4.5-4.6, and 5.1 Due January 30

The first two sections seemed relatively similar to what has been covered in earlier chapters.  It was simply taking properties that we applied to statements P and Q and then having them apply in the case of sets.  5.1 seems easy enough.  You just find one example that disproves the given result and you're finished.  The only difficulty I foresee having is finding out what example to choose.  It seems as if it could be fairly time consuming simply guessing and then checking.

Sec. 7, 4.3-4.4 Due January 27

Section 4.3 was fairly easy and straight forward.  In section 4.4 I failed to understand why the venn diagram would not be able to constitute a proof.  It seemed much simpler.  The hardcore proof works but leads to confusion as it is just a large block of text rather than a few circles with different bits shaded in.
On my homework assignments it takes me roughly two hours to complete the paper homework and an additional half hour for the reading and blogging.  The reading and lecture before the homework has been sufficient up to this point.
The most beneficial thing from the class thus far is seeing all the material twice; once when I read and the second time in class.
The main thing to make the class better for me would be to improve with LaTeX.  The assignment I turned in Wednesday using it took me a ridiculously long time.

Sec. 7, 4.1-4.2 due January 25

The most difficult part was understanding the notation.  I think I've a decent grasp of it, but stressing it in class would go a long way to allowing me to fully understand it.  I'm still struggling to see n | (a-b) as "a is congruent to b modulo n.  Aside from this it seems a fairly straight forward extension of the sections from the previous class period.

Sec. 7, 3.4-3.5 due January 23

Proof by cases seems relatively straight forward.  The only bit that gave me any pause when the book started talking about a proof without loss of generality. This can be used when the proofs necessary to do the proof by cases are fairly similar.  It seems to me that there should be some exception that would make this method no longer functional.  It seems generally safer to do all of the cases, even though it is slightly more time consuming.

Sec. 7, 3.1-3.3 Due January 20th

The most difficult thing for me to follow was the books use of a lemma in a proof.  The proof was to veryify for integers that if 5x-7 were odd, then 9x+2 was even.  The lemma used was that if 5x-7 is odd then x is even.  I failed to understand were they pulled x being even from, and how it was known that this would assist in the proof.  Aside from that I found proofs by contrapositive to be interesting.  They made sense and appear to be useful in many cases.

Sec. 7, 0 (pgs 5-12), due January 18

Chapter 0 was not terribly difficult.  Some of the suggestions I had never though of before (such as not starting a statement with a symbol).  The main difficulty I'm currently having is in installing LaTeX.  This is the focus of Wednesday's class, so hopefully my troubles will be resolved then.

Sec. 7, 2.9-2.10, due January 13

Learning the various fundamental logical equivalencies is both the most difficult portion of the reading, as well as the most useful.  Memorizing them will take time, but when this is accomplished it will mean less and less of truth tables (which while not the most horrible thing, are time consuming and tedious to create).

Sec. 7, 2.5-2.8, due January 11

The most difficult portion of the section was the bit on logical equivalence.  Of everything in the reading thus far this has been the seemingly most convoluted.
The most helpful part of the section was just the author taking the mathematical symbols and writing it as it would appear in plain English.

Sec. 7, 2.1-2.4, Due January 9

From the sections the most difficult thing was simply learning all of the new notation that I had no previous exposure to.
It is fairly cool to be reintroducing such common things as functions, but the different approach gives a deeper understanding of such things.

Sec. 7, 1.1-1.6, due January 6

The most difficult portion of the text was distinguishing between elements and subsets.  At times it became rather confusing as to which was which.  The most useful bit of the sections was in using a Venn diagram in order to evaluate set operations.

Introduction Sec. 7, due on January 6th

I am currently a first year student at BYU.  I have yet to declare a major, but will have math as a minor at the very least.  I've passed Math 112 through AP credit, and have finished Math 113 last semester.  I'm taking the class as it is a requirement for a math minor/major.  The most effective math teacher I've had to date was for AP calculus.  The main thing that worked for me was the shear number of examples and practice problems that were given in order to understand a concept.  As for myself I am currently training for the Salt Lake City Marathon in April.