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.

9.1-9.2, due by February 26, 2014

      The most difficult idea was the one presented in 9.2 but the idea is to produce all possible functions from one set to another.  I can see that this would be possible on finite sets but I do not see how one would start to express this, or even need this for all the sets on an set of infinite size.
      I liked being presented with the different and developing definition of a function.  I can see that the first few differed from ones that followed but I could not quite tell the difference between the last few.  Or at least what the implications of that definition could be.

8.6, due 24 February 2014

      The most difficult part of the reading is knowing if a set is closed under a certain operation.  It seems rather simple but keeping it straight that when elements of a set undergo an operation that the result is in that same set.  I suppose that means that division of integers is not closed as the result could be a rational number.
      I like the idea of addition and multiplication of equivalence classes as it allows us to manipulate and interact with equivalence classes in a more interesting way.  It allows us also to relate equivalence classes to other ones and to do more interesting proofs.

8.5, due February 21, 2014

      The idea of congruency is not new so that was pretty straight forward.  I'm not sure that I could recreate the proof for the symmetric case from memory though.  Remembering how to do the proofs for the different properties of equivalence relations is still a little difficult.
      I liked the significance of the equivalence sets when related to the congruent to modulo relation.  The way to interpret the equivalence sets and determine how many distinct sets should exist, is an interesting idea as well.

8.3-8.4, due February 19, 2014

      I struggled to understand equivalence classes.  I think the difficulty was the new notation without too much notice.  They make sense to me in that when each element is only in one equivalence class and there are no nonempty equivalence classes that a partition is formed.  They also seem to follow rules of sets but I do know know what I am really looking at or what is being communicated to me when reading about equivalence classes.
      I find them interesting as these equivalence classes remind me of topic in linear algebra like vectors and being non-colinear.  It seems to no that understanding these equivalence classes would expand on my understanding of linear algebra.  I also find it exciting that equivalence relations seem to follow patterns when discussing different sets such as lines, number sets and even sets themselves.

8.1-8.2, due February 18, 2014

      The most difficult part to understand was the idea of a relation being transitive.  I did not understand it until I was given examples of specific relations and what it mean for them to be transitive or not.  Being transitive is to say that if a relates to b and b relates to c then a relates to c.  An example of a transitive relation is the divides.  Like if a divides b and b divides c then a divides c for real numbers a, b, c.
      Knowing the properties of certain relations can be quite helpful as one tries to manipulate a given result to prove or disprove it.  One can know useful ideas that we used in induction such as if a > b and b > c then a > c.  But it would be a mistake to say if a is not equal to b and b is not equal to a then a is not equal to a.  Keeping in mind the properties allows for steps to be taken that could not otherwise be taken.

6.4, due by 14 February

      The strong principle of mathematic induction seemed quite simple, though proposing formulae for recursive functions seem a little difficult.  Other than writing out the first few cases and guessing what it might be, then testing it by proof, I'm not sure I see a much better way.  I should be excited to learn what other ways there are.
      This idea continues to expand our ability to use induction.  I like the idea that we can now assume that all elements in a domain from the least element to k can be assumed true in order to prove the implication that the case n = k + 1 is true.  As shown in the book, this will become quite useful when dealing with recursive functions.

6.2, due February 12

      The proof of De Morgan's law for any finite number of sets seemed difficult.  I saw how the proof was done towards the end and it makes sense but the beginning was unclear to me.  I will need to do more practice with these as I prepare for the test.
      I an see that induction is useful as it allows us to now prove results for any well-ordered set.  To be able to induce that something is true over such a large domain is rather exciting.

6.1, due February 10

      I really did not understand the significance of mathematical induction.  It seems to me like for a given sentence you prove that it works for an actual case, then for a generalized case and then for another generalized case proving that it works the same way.  I can see that it could be useful in cases lie that shown in the book about series and sums.  I just do not yet have a clear picture on how it would work.
      I was excited by the example of counting squares.  I have had that problem given to me in the past but I just counted them out.  I am excited now though because I see the generalized case where a square is dimensions n x n, the number of squares is n^2 + (n-1)^2 + (n-2)^2+...+(n-k)^2, where n = k.

Review, due February 7, 2014

      I think that the most important topics are direct proofs, proofs by contrapositive and proofs by contradiction.  It will also be important to review sets as every proof we've done has related in some degree to sets (if only as a domain).

      I expect to see questions about proving implications such as the parity of a number, given another.  I expect to see proofs comparing equality of sets and proofs determining whether something is a rational number given rational and irrational elements of it.

      I need to understand how to evaluate proofs better.  I think it would be helpful to understand expectations for how to evaluate a proof.  Question 5.46 in the book would be helpful to review beforehand.

5.4-5.5, due February 5

      Existence proofs seem rather straight forward.  I found it difficult to intuitively see the methods for beginning existence proofs, especially those relating to rational and irrational numbers.  It took me a little while to understand the example that there exist 2 rational numbers a and b, such that a^b is rational.  I will need to get more comfortable with these proofs, as usual, by experience.
      I enjoy the thought of disproving existence statements.  It requires finding contradictions in the existential case or proving more generalized "for all" statements.  I can see that would begin to be quite a powerful tool.

5.2-5.3, due on February 3

      The proof by contradiction was a little difficult to wrap my mind around.  I found though that it was quite a bit simpler than the formal definition made it sound.  One simply assumes P is true and Q is false or assumes the entire implication is false and then solves for a contradiction.  It makes sense that one would be best off stating that this method will be used for a given result.
      I enjoy adding methods of proofs to my arsenal of proof solving tools.  I can see that the proof by contradiction can be useful in situations where a direct proof or proof by contrapositive is so much more difficult.  This is useful to understand as there must be many situations where this is the simplest method to use.