Review, due April 14, 2014

Which topics and theorems do you think are the most important out of those we have studied?

      I think that the Schroder-Bernstein Theorem.  The idea of implications and biconditionals are probably most important.  I like the fundamental theorem of arithmetic and the division algorithm.  I could see proofs by induction being very useful.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

      I think just getting the calculus proofs down would be useful.  Anything will be helpful there.
What have you learned in this course? How might these things be useful to you in the future?

      I realize to a fuller extent, the importance of being specific and precise in language, particularly in math.  If we don't use precise language, we are prone to mistakes and problems, even if we don't foresee them.  I will be more precise in my language from now on.

12.5, due by April 10, 2014

      Continuity all seems to make sense.  It seems that the only addition here for proofs is to show that the limit exists and that the limit is equal to the value of the function at that point.  I think this section should be most straight forward so long as I can keep delta-epsilon proofs straight.
      This is a useful topic though because it connects limits with the actual function to see it the limit that we predict is included where we think it should be.  I wonder though how we would work off of limits approaching from one side or from another.

12.4, due by April 9, 2014

      Most of the proofs there seemed quite difficult to follow.  I think I need to go back to section 12.3 again to see if I really understand the proofs there.  They still feel a little complicated and I can't seem to keep my variables straight.  I'm sure it will work out after I've done the homework problems for section 12.3.
      This section makes proving the limits of many things so much easier.  I especially liked the proofs at the end of the section which proved generalized limits of polynomials by induction.  It just makes things so much easier.

12.3, due April 7, 2014

      I think delta-epsilon proofs make sense, they just take a little more getting used to with keeping track of the numbers.  As always, I will have to do plenty of practice problems.
      Back in calculus they felt backward for the reason that you can only do them if you have a sense of what the limit is and there never seemed much reason to prove it otherwise.  Though now it makes a lot more sense why we would want to approach limits from a delta-epsilon perspective.

12.2, due by April 4

      The most difficult part will probably be making a good guess for what the pattern is for the formula of the sum.  I will live in hopes that I will not be assigned problems that are too hard.  Otherwise it should work out.
      I enjoy the proofs about series.  I like the idea of making a lemma so that it just becomes a problem of finding the limit.  I find it impressive how there even exist methods to take series and convert them to a formula that can be solved so much easier to deal with.

12.1, due by April 2, 2014

      The most difficult part was keeping straight N, L, and epsilon and what the significance of each one.  When I walk myself through the proofs and keep all of those variables straight, the proof is apparent so I just need practice.
      I like the proofs in calculus.  It is interesting to see that the tools we have to work with can be applied to calculus very easily when given clear definitions.  I like the idea that these proofs can be done in terms of integers and such.

Review, due March 31

• Which topics and theorems do you think are the most important out of those we have studied?
• The Schroder-Bernstein Theorem
• The Fundamental Theorem of Arithmetic
• gcd
• The Division Algorithm
• Cardinality of denumerable and nondenumberable sets
• What kinds of questions do you expect to see on the exam?
• I expect to see a lot of definitions, and proofs
• Specific proofs I expect to see are ones using the Schroder-Bernstein theorem, which should be pretty straight forward.  I hope not to see one where I need create a bijective function that from a subset of the real numbers to the real numbers but I will be prepared.  I expect to see some questions on relative primes and the division algorithm.  It should be a fun exam.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
• I need to understand problems that involve uncountable sets and using the decimal expansion to prove things.  I think 10.36 would help me understand things better.

11.5-6, due March 28

      The proof for the fundamental Theorem of Arithmetic seemed a little complicated.  I will need to review the theorem again before we review it in class.  I followed almost all of it but when dealing with p_1*p_2*p_3*...*p_s = q_1*q_2*q_3*...*q_t.  The part after that was a little complicated.
      I enjoyed reading about the different ways to determine divisibility by 2^n, 9, and 11.  I may have known those in the past but found them interesting and useful as I read about them this time.

11.3-4, due March 26

      Though the proofs are clear for the greatest common divisor, it seems as if the same thing was being stated twice when discussing the 2nd definition of greatest common divisor.  I don't see what the distinguishing part is about the other definition such that it is worth restating differently since we already proved everything to go into that definition.
      I like the Euclidean Algorithm as it is naturally recursive.  I decided that I will write a simple program in C++ to solve for the greatest common divisor.  I am a little frustrated as I do not yet know the conventions for recursion in C++ but I am sure I'll figure it out.

11.1-2, due March 24

      I found it difficult to follow the Division Algorithm proof the first time I read it.  Though being more careful to not the notation, I found it much easier to follow and may be able to replicate it.
      I liked how the division algorithm was related to congruency modulo n.  The division algorithm makes more sense and seems more significant when understanding that connection.

10.5 part 2, due 21 March 2013

      In the proof of Theorem 10.19, it felt as if not every subset was represented in the function f.  Consider for instance {42}.  I do not see how the f(x) ever would result {42}.  I'm not sure what I am missing.  I was also a little confused in the proof for the Schröder-Bernstein Theorem.  I must not be keeping my notation straight.
      I find it interesting that the cardinality of the real numbers is numerically equivalent to the cardinality of the power set of the natural numbers.  I thought that Corollary 10.20 was a nice thought too.

10.5 Part 1, Due 19 March 2014

      I did not understand the proof to Theorem 10.17.  With the introduction of new sets and other information, I lost track of what was going on.  The theorem makes sense but the proof is still a little fuzzy.
      I like the idea of mapping the elements of one set to a subset of that same set.  It starts to make comparing cardinalities more exciting.  I am looking forward to seeing what ideas we can take from this idea.

10.4, due March 17, 2014

      I did understand the proof for Theorem 10.15.  It makes sense by intuition but the proof itself seemed unclear because we defined a function f_S and f_T.  I do not understand yet why.
      I liked the idea that we have proven that there is no largest set.  I also found the Continuum Hypothesis interesting but I'm not sure I understand its significance.  Only that it is an interesting idea, that no set has a cardinality that falls between the the cardinality of the natural numbers and the real numbers.

10.3, due by March 14, 2014

      The most difficult part was the proof strategy when the quadratic equation was used.  It didn't quite make sense why they chose the sign they did.  Suppose it was because the "-" did not fall into the correct range where the "+" sign did.
      I like that we are starting to get into visual representations of the math that we are doing.  It helps me to better grasp what is going on in the section.  I find it exciting to be able to start making more functions like that.

10.2, due by March 12

Though we went over it in class the proof to find that the subset of a denumerable set is denumerable.  I just need to read it a little more carefully.
I found it very interesting that the set of all rational numbers is denumerable but I suppose it makes sense considering it is a relation on the integers and that is a subset of the cartesian product of the integers, which is denumerable and a subset of a denumerable set is denumerable.  It seems this denumerable thing is starting to make a lot more sense.

10.1, due March 10, 2014

      The most difficult part is understanding what was being discussed at the beginning about Galileo. I am not sure I understand the infinite sets and proper subsets of it but I am sure I will by the end of the chapter.
      I thought it was interesting that we have not yet defined cardinality of infinite sets but we have started to get into equivalence of infinite sets and talking about equivalence classes.  I think that could be interesting.

Review, March 7, 2014

• Which topics and theorems do you think are the most important out of those we have studied?

      The topic of proof by induction will be very important to know.  Also understanding equivalence relations will be useful.  Also proving in the generalized case that the composition of functions is onto or one-to-one given certain properties of the functions that compose it.

• What kinds of questions do you expect to see on the exam?

I expect to see true of false questions related to whether or not a function has properties of bijective, injective, surjective, transitive, symmetric, reflexive, etc.  I expect to see questions about proving by induction a formula that maps a sum and mostly things we have covered so far.

• What do you need to work on understanding better before the exam? 

      I do not yet understand how to do proofs with equivalence classes.  I am not sure what is sufficient for those proofs and for proof of compositions of function.

• Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I would like to see Number 22 on the practice midterm 3 worked out.

9.6-9.7, due March 5, 2014

      The most difficult part of this will be keeping straight the definition we have established for the inverse function.  It still follows that you just swap the elements of the ordered pairs of a function but to keep the formal definition in sight will certainly be the challenge.
      I like that the composition of two inverse functions results in the identity function.  It makes senses when I go back and think of how I learned it in high school but it is rather exciting to learn it in this newer context of sets, cartesian products and such.

9.5, due by March 3

      I am still not comfortable proving that functions are surjective or injective.  Sitting down and working through proofs will be the way to learn it better.  Otherwise the material seems pretty straight forward.
      The idea of the composition of functions is so important for calculus and in defining what taking the image of of an image is.  I remember using that idea for chain and quotient rules for calculus.  It is such a useful topic.

9.3-9.4, due 28 February 2014

      I did not understand the idea with bijective functions of equivalence classes.  I do not clearly understand how the proofs work and why we can just switch to the namesake element of the equivalence class when the proof wants to prove it on the entire equivalence class.  If I understood that better, I think the proofs with them would make more sense.
      I like the idea of injective and surjective functions.  In high school I remember discussing one-to-one functions but I remember wondering why we did not discuss onto functions and if they had any significance.  I like to the thought that they are worth discussing but I am looking forward to see why.