4.4-4.5,5.1, due January 31

      The proofs involving Cartesian products of sets seemed a little more difficult.  It seems as if keeping all the different elements straight will be quite difficult.  However, the idea is fairly straight forward.  One need take the variables separately and remember what sets the individuals are elements of.  Of course, with practice, it will continue to become more clear.
      I find the counterexamples to be very useful.  I was in a debate earlier today, the opposing side used a generalization in a certain domain.  I thought of a counterexample that disproved their statement.  Though the counterexample did not prove their whole argument as false (because debates are much more subjective than mathematics) I felt logically validated for my use of a counterexample.

4.3-4.4, due January 29

      I found it difficult to understand the proof that verified a particular equality or inequality.  In the example, Result 4.16, I did not understand why (2x - 3y) squared is greater than or equal to zero was used.  I then realized that it came from working the result backwards until the equality could be shown in a way that was known to be true.  Because (2x - 3y) squared is always greater than or equal to zero, it is a great starting point for a direct proof.  It is now much more clear how to verify an equality or inequality.
      I found it interesting to begin working with proofs within sets because it allows us to solve much more general proofs involving sets of numbers rather than just individual number.  I like that we are expanding the tools that we are using to prove statements and expanding the situations in which those tools can be used.

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I have typically spent half an hour to three quarters of an hour on reading and another hour on homework problems.  When I wrote my second assignment in LaTeX, I spent about three hours completing it.  I do not anticipate them taking so long as I improve my skills in LaTeX and as I modify the template so that I do not have to redo so much in the assignments every time.
• What has contributed most to your learning in this class thus far?
• I have found that writing about the most difficult part of the reading has helped me to synthesize the material better.
• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
• I think I would learn more if I could see where this will be applied in the various topics within math.  I sure as we continue through the year it will become clearer.

4.1-4.2, due January 27, 2014

      The most difficult part of the reading was understanding the proofs involving congruence of integers.  Once I understood what the "congruent to... modulo" statement meant mathematically, I realized how simple the proofs were but until that point, it did not make much sense.
      I was rather excited when seeing the statements about how for a given integer x, 2 divides (x - 0) or 2 divides (x -1).  Also 3 divides (x - 0), 3 divides (x - 1) or 3 divides (x - 2).  It seemed exciting because for the general case of an integer n, n divides (x - 0), n divides (x - 1) ... n divides (x - (n - 1)).  This could be applied when dealing with proofs involving sets.  I am excited to deal with proofs of more general cases.

3.3-3.5, due January 24

      The most difficult part was understanding proof by contrapositive.  I understand that "if P, then Q" is logically equivalent to "if not Q, then not P."  The confusion come when I try to keep straight which statement I am trying to prove true with a given result.  I am sure that I will understand it better with practice.
      I am impressed by section 3.5 on proof evaluation.  It was not until then that I realized that one proof could prove a variety of results so long as the results are logically equivalent.  Keeping this idea in mind could be particularly useful when employing different techniques to prove results.  It is also useful to be able to think critically about a given proof to determine whether it is a convincing proof or not.

3.1-3.2, due January 22

      The reading on proofs was interesting.  The most difficult part was when I skipped a page explaining the proof method to prove an integer is positive or negative.  Going back, realizing I missed a page, I read the idea.  It took me a few readings to realize what they were trying to say.  For an even number, the method is to show the formula can be written to as an arbitrary integer, multiplied by 2.  For an odd integer, it is to show an integer being multiplied by 2 then adding 1.  Understanding that made the following proofs make much more sense.
      I find the method shown to prove that an integer is positive or negative is very interesting.  I'm not sure how many significant proofs there are that would require this but it is good to understand the logic behind it.  This idea has expanded my knowledge on the functionality of substitution because in the end the part that is an integer could be expressed as a single variable.

Chapter 0, due January 17

      The most difficult part of the reading was determining whether to use "that" or "which" in specific cases.  That is used to introduce a restrictive clause.  This is to specify the exact object being referred to.  Which is used for nonrestrictive clauses.  It seems to be used for giving more information about something.  It is for unnecessary or additional information.
      I thought it was very useful to have the conventions used in mathematical writing expressed clearly.  I had never thought before that it might be confusing to start a sentence with a symbol.  Now that I've read about good mathematical writing, I can be more conscientious about that which I write in this class and in future classes.

2.9-2.10, due on January 15

      This reading was fairly straight forward.  I did not understand why the negation of a quantified statement was being discussed and why during the negation, the inverted "A" was changed to the leftward-pointing "E."  I took a step back and really tried to understand the notation and it became clear that "if all elements x of set S for P(x)" is an statement then the negation would be there exists an element x that is an element of ~P(x) must be the negation.  It is now clear in my mind.
      I was interested by the notation for quantified statements as I often look up information on wikipedia and now I understand what more of the notation means.  This should help me when needing quick information related to math.