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.